Vector Calculus in maths is a sub-division of Calculus that deals with the differentiation and integration of Vector Functions. We already know that Calculus is a branch of mathematics that deals with the rate of change of a function concerning another function. There are two major divisions of Calculus namely, Differential Calculus and Integral Calculus.
The branch of Differential Calculus deals with the process of finding derivatives or differentiation of functions while Integral Calculus deals with finding the antiderivative of a function whose derivative is given. In this article, we will learn in detail about Vector Calculus which is a lesser-known branch of calculus, and the basic formulas of Vector Calculus.
In this article, you are going to read everything about what is vector calculus in engineering mathematics, vector calculus formulas, vector analysis, etc.
What is Vector Calculus?
Vector Calculus is a branch of mathematics that deals with the operations of calculus i.e. differentiation and integration of vector field usually in a 3 Dimensional physical space also called Euclidean Space. The applicability of Vector calculus is extended to partial differentiation and multiple integration. Vector Field refers to a point in space that has magnitude and direction. These Vector Fields are nothing but Vector Functions. Vector calculus is also known as vector analysis.
The vector fields are the vector functions whose domain and range are not dimensionally related to each other. The branch of Vector Calculus corresponds to the multivariable calculus which deals with partial differentiation and multiple integration. This differentiation and integration of vector is done for a quantity in 3D physical space represented as R3. For n-dimensional space, it is represented as Rn.
Read in Detail: Calculus in Maths
Vector Calculus Definition
Vector calculus, also known as vector analysis or vector differential calculus, is a branch of mathematics that deals with vector fields and the differentiation and integration of vector functions
Vector Calculus often called Vector Analysis deals with vector quantities i.e. the quantities that have both magnitude as well as direction. Since we know that Vector Calculus deals with differentiation and integration of functions, there are three types of integrals dealt with in Vector Calculus that areÂ
- Line Integral
- Surface Integral
- Volume Integral
Let's learn about these integrals in detail.
Line Integral
Line Integral in mathematics is the integration of a function along the line of the curve. The function can be a scalar or vector whose line integral is given by summing up the values of the field at all points on a curve weighted by some scalar function on the curve. Line Integral is also called Path Integral and is represented by Φ = ∫Lf. Line Integral has got its application in physics. For Example, Work Done by Force is along a path given as W = ∫LF(s).ds because we know that work done is given as the product of force and distance covered.
Surface Integral
Surface Integral in mathematics is the integration of a function along the whole region or space that is not flat. In Surface integral, the surfaces are assumed of small points hence, the integration is given by summing up all the small points present on the surface. The surface integral is equivalent to the double integration of a line integral. Surface Integral has got its application in Electromagnetism and many more branches of physics where the vector function is spread over the surface. Surface Integral is represented as ∬sf(x,y)dA.
Learn more about Double Integral.
Volume Integral
A volume integral, also known as a triple integral, is a mathematical concept used in calculus and vector calculus to calculate the volume of a three-dimensional region within a space. It is an extension of the concept of a definite integral in one dimension to three dimensions.
Mathematically, the volume integral of a scalar function f(x, y, z) over a region R in three-dimensional space is denoted as:
\bold{\iiint_R f(x, y, z) \ dV}
WhereÂ
- dV represents an infinitesimal volume element, andÂ
- Integral is taken over region R.
Operation in Vector
The different operations performed with vector quantities are tabulated below with their notation and illustration.
| Operation | Notation | Illustration |
|---|
Vector Addition | r1 + r2 | Addition of two vectors gives a vector |
Scalar Multiplication | q.r1 | Multiplying a vector 'r1' with scalar 'q' result in a vector |
Dot Product | r1 · r2 | Dot product of two vectors gives a scalar |
Cross Product | r1 ⨯ r2 | Cross product of two vectors gives a vector |
Scalar Triple Product | r1 · (r2 ⨯ r3) | Dot Product of Cross product of two vectors |
Vector Triple Product | r1 ⨯ (r2 ⨯ r3) | Cross Product of Cross Product of two Vectors |
Learn More, Dot and Cross Product of Vectors
Divergence and Curl
Divergence and Curl are two important operators used in Vector Calculus. Divergence is a scalar operator which tells about the behaviour of a function towards or away from a point. Curl is a vector operator which tells about the behaviour of a function around a point. The vector operator is represented by ∇ which accounts for the partial differentiation of the vector field. The Vector Differential Operator (∇) also called Nabla is expressed as ∇ = ∂/∂x i + ∂/∂y j + ∂/∂z k.
Divergence of Vector
If a vector field is given by f(x,y,z) = fxi + fyj + fzk then its divergence is given by taking the scalar of the vector operator is given by
div(f) = ∇.f(x,y,z) = (∂/∂x i + ∂/∂y j + ∂/∂z k) · (fxi + fyj + fzk )
⇒ ∇.f(x,y,z) = ∂x/∂x + ∂y/∂y + ∂z/∂z.
Curl of Vector
If a vector field is given by f(x,y,z) = fxi + fyj + fzk then its curl is given by taking the vector of the vector operator
∇ × f(x,y,z) = (∂/∂x i + ∂/∂y j + ∂/∂z k) ⨯ (fxi + fyj + fzk )
⇒ ∇ × f(x,y,z) = \begin{vmatrix} i & j & k \\ \partial /\partial x& \partial /\partial y & \partial /\partial z \\ f_{x} & f_{y} & f_{z} \\ \end{vmatrix}
⇒ ∇ × f(x,y,z) = (∂z/∂y - ∂y/∂z)i + (∂x/∂z - ∂z/∂x)j + (∂y/∂x - ∂x/∂y).
Gradient of Scalar
The gradient of a scalar field F is given by grad(F) or ∇ F. It gives the measurement of the rate and direction of a scalar-valued function. In the Cartesian system, the gradient of a scalar-valued function is given by
∇ F = (∂/∂x i + ∂/∂y j + ∂/∂z k)F = ∂/∂x i + ∂/∂y j + ∂/∂z k
For a vector field given as F(x,y,z) = p(x,y,z)i + q(x,y,z)j + r(x,y,z)k. The following formulas are given.
Fundamental Theorem of Line Integral
if F = ∇Φ and Curve C has A and B endpoints then its line integral is given as
∫cF.dr = Φ(B) - Φ(A)
There are two theorems under Circulation Curl Form, namely the Green theorem and Stokes theorem.
Green Theorem: If D is the region bounded by curve C then, ∮cF.dr = ∬D(∂Q/∂x - ∂P/∂y)dA
Stoke's Theorem: For a surface S bounded by curve C stokes theorem given by ∮cF.dr = ∬S(∇ ⨯ F)dS
Flux Divergence Theorem
The Flux Divergence Form of Green's Theorem is given as ∬D∇. F dA = ∮cF.n ds
The Flux Divergence Form of Stoke's Theorem is given as âˆD ∇. F dV = ∯sF.n dσ
Vector Calculus Identities
The list of Vector Calculus Identities have been tabulated under three categories.
Gradient Function Identities
The Gradient Function Identities are tabulated below:
\vec{\nabla}Â Â Â Â
(f + g) | \vec{\nabla}Â Â Â Â
f +Â \vec{\nabla}Â Â Â Â
g |
|---|
\vec{\nabla}Â Â Â Â
(cf) | c\vec{\nabla}Â Â Â Â
f, where C is constant |
|---|
\vec{\nabla}Â Â Â Â
(fg) | f\vec{\nabla}Â Â Â Â
g + g\vec{\nabla}Â Â Â Â
f |
|---|
\vec{\nabla}Â Â Â Â
(f/g) | (g\vec{\nabla}Â Â Â Â
f - f\vec{\nabla}g   Â
)/g2 at point \vec{x}   Â
where g(\vec{x}Â Â Â Â
) ≠0 |
|---|
\vec{\nabla}(\vec{F}+\vec{G})
| \vec{F} \times(\vec{\nabla} \times \vec{G})-(\vec{\nabla} \times \vec{F}) \times \vec{G}+(\vec{G} \cdot \vec{\nabla}) \vec{F}+(\vec{F} \cdot \vec{\nabla}
|
|---|
Divergence Function Identities
The identity formula for Divergence Function is tabulated below:
\vec{\nabla}(\vec{F}+\vec{G})
| \vec{\nabla} \cdot \vec{F}+\vec{\nabla} \cdot \vec{G}
|
|---|
\vec{\nabla} \cdot(c \vec{F})
| c \vec{\nabla} \cdot \vec{F}
|
|---|
\vec{\nabla} \cdot(f \vec{F})
| f \vec{\nabla} \cdot \vec{F}+\vec{F} \cdot \vec{\nabla}
|
|---|
\vec{\nabla} \cdot(\vec{F} \times \vec{G})
| \vec{G} \cdot(\nabla \times \vec{F})-\vec{F} \cdot(\nabla \times \vec{G})
|
|---|
Curl Function Identities
The identities for curl function is tabulated below:
\vec{\nabla} \times(\vec{F}+\vec{G})
| \nabla \times \vec{F}+\vec{\nabla} \times \vec{G}
|
|---|
\vec{\nabla} \times(c \vec{F})
| c \vec{\nabla} \times \vec{F}
|
|---|
\vec{\nabla} \times(f \vec{F})
| f \vec{\nabla} \times \vec{F}+\vec{\nabla} f \times \vec{F}
|
|---|
\vec{\nabla} \times(\vec{F} \times \vec{G})
| \vec{F} \cdot(\vec{\nabla} \cdot \vec{G})-(\vec{\nabla} \vec{F}) \vec{G}+(\vec{G} \cdot \vec{\nabla}) \vec{F}-(\vec{F} \cdot \vec{\nabla})
|
|---|
Laplacian Function Identities
The identities for Laplacian Function is tabulated below:
\vec{\nabla}^2(f+g)
| \vec{\nabla}^2 f+\vec{\nabla}^2 g
|
|---|
\vec{\nabla}^2(c f)
| c \vec{\nabla}^2 f   Â
, where C is a constant |
|---|
\vec{\nabla}^2(f g)
| f \vec{\nabla}^2 g+2 \vec{\nabla} f \cdot g+g \vec{\nabla}^2
|
|---|
Degree Two Function Identities
The vector calculus identities for two degree function is tabulated below:
\vec{\nabla} \cdot(\nabla \times \vec{F})
| 0 |
|---|
\vec{\nabla} \times(\vec{\nabla} f)
| 0 |
|---|
\vec{\nabla} \cdot(\overrightarrow{\nabla f} \times \overrightarrow{\nabla g})
| 0 |
|---|
\vec{\nabla} \cdot(f \overrightarrow{\nabla g}-g \vec{\nabla})
| f \vec{\nabla}^2 g-g \vec{\nabla}^2 f
|
|---|
\vec{\nabla} \times(\nabla \times \vec{F})
| \vec{\nabla}(\vec{\nabla} \cdot \vec{F})-\vec{\nabla}^2
|
|---|
Vector Calculus Applications
Vector Calculus or vector analysis has a number of applications in the real world:
Related Resources,
Solved Examples on Vector Calculus
Example 1: If F(x,y,z) = 3xy2 - y2z3 then find gradF or ∇ f.
Solution:
∇ f = (∂/∂x i + ∂/∂y j + ∂/∂z k)(3xy2 - y2z3)
⇒ ∇ f = ∂/∂x(3xy2 - y2z3)i + ∂/∂y(3xy2 - y2z3)j + ∂/∂z (3xy2 - y2z3)k
⇒ ∇ f = 3y2i + (6xy - 2yz3) + (-3y2z2)k
Example 2: Find the div(F) or ∇·F, if F = xz2 i - 2y2z3 j + x2yz k
Solution:
div(F) = ∇·F = (∂/∂x i + ∂/∂y j + ∂/∂z k)·(xz2 i - 2y2z3 j + x2yz k)
⇒ ∇·F = ∂/∂x(xz2) + ∂/∂y(- 2y2z3) + ∂/∂z(x2yz)
⇒ ∇·F = z2 - 4yz3 + x2y
Example 3: Find curl F i.e. ∇ × f if F = xz2 i - 2x3yz j + 2yz4 k
Solution:
∇ × f= (∂/∂x i + ∂/∂y j + ∂/∂z k) ⨯ (xz2 i - 2x3yz j + 2yz4 k)
⇒ ∇ × f = \begin{vmatrix} i & j & k \\ \partial /\partial x& \partial /\partial y & \partial /\partial z \\ xz^{2} & 2x^{3}yz & 2yz^{4} \\ \end{vmatrix}
⇒ ∇ × f = [∂/∂y(2yz4) - ∂/∂z(-2x3yz)]i + [∂/∂z(xz2) - ∂/∂x(2yz4)]j + [∂/∂x(-2x3yz) - ∂/∂y(xz2)]k
⇒ ∇ × f = [2z4 + 2x3y]i + [2xz - 0]j + [-6x2yz - 0]k
⇒ ∇ × f = (2z4 + 2x3y)i + (2xz)j -(6x2yz)k
Solved Examples
1. Calculate the gradient of f(x,y,z) = x^2y + yz^3 + xz
Solution:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
∂f/∂x = 2xy + z
∂f/∂y = x^2 + z^3
∂f/∂z = 3yz^2 + x
∇f = (2xy + z, x^2 + z^3, 3yz^2 + x)
2. Find the divergence of F(x,y,z) = (x^2 + y)i + (y^2 + z)j + (z^2 + x)k
Solution:
div F = ∇ · F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z
∂F_x/∂x = 2x
∂F_y/∂y = 2y
∂F_z/∂z = 2z
div F = 2x + 2y + 2z
3. Calculate the curl of F(x,y,z) = (yz)i + (xz)j + (xy)k
Solution:
curl F = ∇ × F = (∂F_z/∂y - ∂F_y/∂z)i + (∂F_x/∂z - ∂F_z/∂x)j + (∂F_y/∂x - ∂F_x/∂y)k
∂F_z/∂y - ∂F_y/∂z = x - x = 0
∂F_x/∂z - ∂F_z/∂x = y - y = 0
∂F_y/∂x - ∂F_x/∂y = z - z = 0
curl F = 0i + 0j + 0k = 0
4. Evaluate ∫_C (x^2 + y^2) ds, where C is the circle x^2 + y^2 = 1, traversed counterclockwise.
Solution:
Parameterize the circle: x = cos(t), y = sin(t), 0 ≤ t ≤ 2π
ds = √((dx/dt)^2 + (dy/dt)^2) dt = √(sin^2(t) + cos^2(t)) dt = dt
∫_C (x^2 + y^2) ds = ∫_0^(2π) (cos^2(t) + sin^2(t)) dt = ∫_0^(2π) 1 dt = 2π
5. Compute the flux of F(x,y,z) = xi + yj + zk through the surface of the sphere x^2 + y^2 + z^2 = a^2.
Solution:
Use the divergence theorem: ∫∫_S F · dS = ∫∫∫_V div F dV
div F = 1 + 1 + 1 = 3
Volume of sphere = (4/3)Ï€a^3
Flux = ∫∫∫_V 3 dV = 3 * (4/3)πa^3 = 4πa^3
Practice Problems on Vector Calculus in Maths
1. Calculate the gradient of f(x,y,z) = ln(x2 + y2 + z2)
2. Find the divergence of F(x,y,z) = (x2y)i + (y2z)j + (z2x)k
3. Compute the curl of F(x,y,z) = (y2)i + (z2)j + (x2)k
4. .Evaluate the line integral ∫_C (x + y) ds along the curve C: x = t, y = t2, z = t3, 0 ≤ t ≤ 1
5. Calculate the surface integral of F(x,y,z) = xi + yj + zk over the surface z = x2 + y2, 0 ≤ z ≤ 1
6. Find the gradient of f(x,y) = e(x+y) sin(xy)
7. Determine if F(x,y,z) = (yz)i + (zx)j + (xy)k is conservative
8. Compute the divergence of F(r,θ,φ) = r^ cos(φ)r + r sin(θ) sin(φ)θ + r cos(θ)φ in spherical coordinates
9. Find the potential function for F(x,y) = (2x + y)i + (x + 2y)j
10. Calculate the work done by F(x,y,z) = xi + yj + zk along the helix r(t) = (cos(t), sin(t), t), 0 ≤ t ≤ 2π
Summary
Vector calculus is a branch of mathematics that deals with vector fields in two or more dimensions. It encompasses key concepts such as the gradient, which measures the rate and direction of change in scalar fields; divergence, which quantifies the "outflow" of vector fields; and curl, which describes the rotation of vector fields. The field also includes techniques for integration over curves (line integrals) and surfaces (surface integrals). Fundamental theorems like the Divergence Theorem and Stokes' Theorem connect these concepts, while the study of conservative fields and potential functions provides insights into path independence and field representations. These tools and concepts are essential for analyzing complex systems in physics, engineering, and other scientific disciplines, allowing for the mathematical description of phenomena involving fluid dynamics, electromagnetism, and other multidimensional processes.
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Mean Value TheoremThe Mean Value Theorem states that for a curve passing through two given points there exists at least one point on the curve where the tangent is parallel to the secant passing through the two given points. Mean Value Theorem is abbreviated as MVT. This theorem was first proposed by an Indian Mathem
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Rolle's Mean Value TheoremRolle's theorem one of the core theorem of calculus states that, for a differentiable function that attains equal values at two distinct points then it must have at least one fixed point somewhere between them where the first derivative of the function is zero.Rolle's Theorem and the Mean Value Theo
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Lagrange's Mean Value TheoremLagrange's Mean Value Theorem (LMVT) is a fundamental result in differential calculus, providing a formalized way to understand the behavior of differentiable functions. This theorem generalizes Rolle's Theorem and has significant applications in various fields of engineering, physics, and applied m
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Rolle's Theorem and Lagrange's Mean Value TheoremRolle's Theorem and Lagrange's Mean Value Theorem: Mean Value Theorems (MVT) are the basic theorems used in mathematics. They are used to solve various types of problems in Mathematics. Mean Value Theorem is also called Lagrenges's Mean Value Theorem. Rolle’s Theorem is a subcase of the mean value t
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Advanced DifferentiationDerivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
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Tangents and NormalsTangent and Normals are the lines that are used to define various properties of the curves. We define tangent as the line which touches the circle only at one point and normal is the line that is perpendicular to the tangent at the point of tangency. Any tangent of the curve passing through the poin
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Equation of Tangents and NormalsDerivatives are used to find rate of change of a function with respect to variables. To find rate of change of function with respect to a variable differentiating it with respect to that variable is required. Rate of change of function y = f(x) with respect to x is defined by dy/dx or f'(x). For exa
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Increasing and Decreasing FunctionsIncreasing and decreasing functions refer to the behavior of a function's graph as you move from left to right along the x-axis. A function is considered increasing if for any two values x1 and x2​ such that x1 < x2 ​, the function value at x1​ is less than the function value at x2​ (i.e., f( x1)
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Increasing and Decreasing IntervalsIncreasing and decreasing intervals are the intervals of real numbers in which real-valued functions are increasing and decreasing respectively. Derivatives are a way of measuring the rate of change of a variable.Increasing and Decreasing IntervalsWhen it comes to functions and calculus, derivatives
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Concave FunctionGraphs of the functions give us a lot of information about the nature of the function, the trends, and the critical points like maxima and minima of the function. Derivatives allow us to mathematically analyze these functions and their sign can give us information about the maximum and minimum of th
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Inflection PointInflection Point describes a point where the curvature of a curve changes direction. It represents the transition from a concave to a convex shape or vice versa. Let's learn about Inflection Points in detail, including Concavity of Function and solved examples. Table of Content Inflection Point Defi
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Curve SketchingCurve Sketching as its name suggests helps us sketch the approximate graph of any given function which can further help us visualize the shape and behavior of a function graphically. Curve sketching isn't any sure-shot algorithm that after application spits out the graph of any desired function but
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Absolute Minima and MaximaAbsolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the funct
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Relative Minima and MaximaRelative maxima and minima are the points defined in any function such that at these points the value of the function is either maximum or minimum in their neighborhood. Relative maxima and minima depend on their neighborhood point and are calculated accordingly. We find the relative maxima and mini
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Integrals
Indefinite Integration
Indefinite IntegralsIntegrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different
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Integration FormulasIntegration Formulas are the basic formulas used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are beneficial for finding
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Integration by Substitution MethodIntegration by substitution or u-substitution is a highly used method of finding the integration of a complex function by reducing it to a simpler function and then finding its integration. Suppose we have to find the integration of f(x) where the direct integration of f(x) is not possible. So we su
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Integration by Partial FractionsIntegration by Partial Fractions is one of the methods of integration, which is used to find the integral of the rational functions. In Partial Fraction decomposition, an improper-looking rational function is decomposed into the sum of various proper rational functions.If f(x) and g(x) are polynomia
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Partial Fraction ExpansionIf f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime
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Integration by PartsIntegration is the calculation of an integral which is used in maths for finding many useful quantities such as areas, volumes, displacement, etc., that occur due to a collection of small data which cannot be measured singularly. Integrals are denoted using " ∫ ".Integration NotationThe concept of i
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Integration by U-substitutionFinding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind
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Reverse Chain RuleIntegrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve
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Trigonometric Substitution: Method, Formula and Solved ExamplesTrigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the
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Integration of Trigonometric FunctionsIntegration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.Below is an example of the Integra
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Definite Integration
Definite Integral | MathematicsDefinite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits. \int_{a}^{b}F(x)dx , It denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit. Note
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Computing Definite IntegralsIntegrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface
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Fundamental Theorem of Calculus | Part 1, Part 2Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for
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Finding Derivative with Fundamental Theorem of CalculusIntegrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit
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Evaluating Definite IntegralsIntegration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function
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Properties of Definite IntegralsProperties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as \int_{a}^{b}f(x) = F(b) − F(a)There are many properties regarding definite integral. We will discuss each property one by one with proof.Defin
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Definite Integrals of Piecewise FunctionsImagine a graph with a function drawn on it, it can be a straight line or a curve, or anything as long as it is a function. Now, this is just one function on the graph. Can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight lin
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Improper IntegralsImproper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b
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Riemann SumsRiemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int
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Riemann Sums in Summation NotationRiemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These
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Definite Integral as the Limit of a Riemann SumDefinite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li
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Trapezoidal RuleThe Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form b∫a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as
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