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Advanced Calculus Problems

This document contains a question bank with multiple choice and numerical problems involving double and triple integrals related to engineering mathematics. Some of the problems involve evaluating double integrals over specific regions, finding directional derivatives, and applying concepts like Green's theorem, divergence, curl and conservative vector fields. The problems cover topics like double and triple integrals, vector calculus, line integrals and surface integrals.

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Agrim Agarwal
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165 views3 pages

Advanced Calculus Problems

This document contains a question bank with multiple choice and numerical problems involving double and triple integrals related to engineering mathematics. Some of the problems involve evaluating double integrals over specific regions, finding directional derivatives, and applying concepts like Green's theorem, divergence, curl and conservative vector fields. The problems cover topics like double and triple integrals, vector calculus, line integrals and surface integrals.

Uploaded by

Agrim Agarwal
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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QUESTION BANK

UNIT-III
ENGINEERING MATHEMATICS-I

1. Evaluate the integrals.


𝑦
1 𝑥 1 𝑥^2
a. ∫0 ∫0 𝑥𝑦 𝑑𝑥 𝑑𝑦 b. ∫0 ∫0 𝑒 𝑥 𝑑𝑥 𝑑𝑦

2. Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦 where 𝐴 is the domain bounded by x-axis, 𝑥 = 2𝑎


2
and 𝑥 = 4𝑎𝑦.

3. Find the area between the parabolas 𝑦 2 = 4𝑎𝑥 and 𝑥 2 = 4𝑎𝑦.

4. Evaluate ∫ ∫ 𝑦 𝑑𝑥 𝑑𝑦 where 𝑅 is the region bounded by the circle 𝑥 2 + 𝑦 2 =


𝑎2 where 𝑥 ≥ 0, 𝑦 ≥ 0.

5. Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦 over the region in first quadrant for which 𝑥 + 𝑦 ≤ 1

6. Evaluate the integrals,

a. ∫ ∫ ∫ (𝑥 + 𝑦 + 𝑧)𝑑𝑥 𝑑𝑦 𝑑𝑧 where 𝑅: 0 ≤ 𝑥 ≤ 1, 1 ≤ 𝑦 ≤ 2, 2 ≤ 𝑧 ≤ 3.

b. ∫ ∫ ∫ (𝑥 − 2𝑦 + 𝑧)𝑑𝑥 𝑑𝑦 𝑑𝑧 where 𝑅: 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 𝑥 2 , 0 ≤ 𝑧 ≤ 𝑥 +
𝑦.

4 𝑥 𝑥+𝑦
c. ∫0 ∫0 ∫0 𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥

7. Evaluate the integrals using Beta and Gamma Functions.


𝜋 𝜋
a. ∫02 sin2 𝜃 cos 4 𝜃 𝑑𝜃 b. ∫02 sin3 𝜃 cos 5 𝜃 𝑑𝜃

8. Evaluate
∞ 2 ∞ 3
a. ∫0 √𝑥𝑒 −𝑥 𝑑𝑥 b. ∫0 𝑒 −𝑥 𝑑𝑥

1
1
9. Express ∫0 𝑥 3 (1 − 𝑥 2 )2 𝑑𝑥.

10. Evaluate the Dirichlet Integrals,


a. 𝐼 = ∫ ∫ ∫ 𝑥 3 𝑦 3 𝑧 3 𝑑𝑥 𝑑𝑦 𝑑𝑧, where T is the region in the first octant bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1 and coordinate plane.
b. 𝐼 = ∫ ∫ ∫ 𝑥𝑦 2 𝑧 3 𝑑𝑥 𝑑𝑦 𝑑𝑧, T is the region bounded by 𝑥 + 𝑦 + 𝑧 = 1 and the
coordinate plane.

c. 𝐼 = ∫ ∫ ∫ 𝑥 2 𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑧, T is the region bounded by 𝑥 2 + 𝑦 2 + 𝑧 2 = 1 and the


coordinate plane.

QUESTION BANK
UNIT-IV
ENGINEERING MATHEMATICS-I
1
1. Find the directional derivative of 𝜙 = (𝑥 2 + 𝑦 2 + 𝑧 2 )2 at the point 𝑃(3, 1, 2) in the
direction of the vector 𝑦𝑥 𝑖̂ + 𝑧𝑥 𝑗̂ + 𝑥𝑦 𝑘̂.

2. Find the directional derivative of 𝑓(𝑥, 𝑦, 𝑧) = 𝑒 2𝑥 𝑐𝑜𝑠𝑦 𝑧 at (0, 0, 0) in the direction


𝜋
of the tangent to the curve 𝑥 = 𝑎 𝑠𝑖𝑛𝑡, 𝑦 = 𝑎 𝑐𝑜𝑠𝑡, 𝑧 = 𝑎𝑡 at 𝑡 = .
4

3. ⃗ = 𝑥 2 𝑦 2 𝑖̂ + 2𝑥𝑦 𝑗̂ + (𝑦 2 − 𝑥𝑦) 𝑘̂.


Find the divergence and curl of the vector field 𝑉

4. Find the constants 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 so that 𝐹 = (𝑥 + 2𝑦 + 𝑎𝑧) 𝑖̂ + (𝑏𝑥 − 3𝑦 − 𝑧) 𝑗̂ +


(4𝑥 + 𝑐𝑦 + 2𝑧) 𝑘̂ is irrotational.

5. Prove that (𝑦 2 − 𝑧 2 + 3𝑦𝑧 − 2𝑥) 𝑖̂ + (3𝑥𝑧 + 2𝑥𝑦) 𝑗̂ + (3𝑥𝑦 − 2𝑥𝑧 + 2𝑧) 𝑘̂ is


both Solenoidal and irrotational.

6. Use Green’s theorem to evaluate ∫𝐶 (𝑥 2 + 𝑥𝑦)𝑑𝑥 + (𝑥 2 + 𝑦ℎ2 )𝑑𝑦 where C is the


square formed by the lines 𝑦 = ±1, 𝑥 = ±1.

7. Using Green’s theorem, evaluate ∫𝐶 (𝑥 2 𝑦 𝑑𝑥 + 𝑥 2 𝑑𝑦) where C is the boundary


described counter clockwise of the triangle with vertices (0, 0). (1, 0), (1, 1).

8. Apply Green’s theorem to evaluate ∫𝐶 (2𝑥 2 − 𝑦 2 )𝑑𝑥 + (𝑥 2 + 𝑦 2 )𝑑𝑦 where C is the


boundary of the area enclosed by the x-axis and upper half of the circle 𝑥 2 + 𝑦 2 =
𝑎2 .
9. Evaluate by Green’s theorem, ∫𝐶 𝑒 −𝑥 𝑠𝑖𝑛𝑦 𝑑𝑥 + 𝑒 −𝑥 cos 𝑦 𝑑𝑦, where C is the
𝜋
rectangle with vertices (0, 0), (𝜋, 0), (𝜋, ) , (0, 𝜋) and hence verify Green’s
2
theorem.

10. Show that the vector field 𝐹 = 2𝑥(𝑦 2 + 𝑥 3 ) 𝑖̂ + 2𝑥 2 𝑦 𝑗̂ + 3𝑥 2 𝑧 2 𝑘̂ is


conservative. Find the scalar potential and work done in moving a particle from
(−1, 2, 1) to (2, 3, 4).

11. Show that the vector field defined by 𝐴 = (𝑥 2 − 𝑦 2 + 𝑥) 𝑖̂ − (2𝑥𝑦 + 𝑦) 𝑗̂ is


irrotational and find the scalar potential.

12. If 𝐹 = 3𝑥𝑦 𝑖̂ − 𝑦 2 𝑗̂. Evaluate ∫𝐶 𝐹. 𝑑𝑟 where C is the arc of parabola 𝑦 = 2𝑥 2 from


(0, 0) to (1, 2).

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