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Integration p1

The document contains multiple word problems involving integration and differentiation. Some key points: 1) The first problem asks students to find the area of a region bounded by a graph, axes, and use it to find the side length of a triangle with the same area. 2) The second problem involves finding the production rate of oil from a well over time using differential equations, and calculating total production over a year. 3) The third problem models the height of a ball over time after being thrown up and finds the time and domain it hits the ground along with the range. 4) Subsequent problems involve using integrals to find the approximate area of a curved theater scenery piece, determining profit growth rates

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Emil Babayev
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28 views11 pages

Integration p1

The document contains multiple word problems involving integration and differentiation. Some key points: 1) The first problem asks students to find the area of a region bounded by a graph, axes, and use it to find the side length of a triangle with the same area. 2) The second problem involves finding the production rate of oil from a well over time using differential equations, and calculating total production over a year. 3) The third problem models the height of a ball over time after being thrown up and finds the time and domain it hits the ground along with the range. 4) Subsequent problems involve using integrals to find the approximate area of a curved theater scenery piece, determining profit growth rates

Uploaded by

Emil Babayev
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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integration [217 marks]

The following diagram shows part of the graph of f(x) = (6 − 3x) (4 + x), x ∈ R.
The shaded region R is bounded by the x-axis, y-axis and the graph of f .

1a. Write down an integral for the area of region R. [2 marks]

1b. Find the area of region R. [1 mark]


1c. The three points A(0, 0) , B(3, 10) and C(a , 0) define the vertices of a [2 marks]
triangle.

Find the value of a , the x-coordinate of C, such that the area of the triangle is
equal to the area of region R.

The production of oil (P ), in barrels per day, from an oil field satisfies the
1000
differential equation ddP = 2+t
where t is measured in days from the start of
t
production.

2a. Find ∫ 5 1000 dt. [1 mark]


0 2+t
2b. State in context what this value represents. [1 mark]

The production of oil at t = 0 is 20, 000 barrels per day.

2c. Find an expression for P in terms of t. [4 marks]

2d. Determine ∫ 365 P (t) dt and state what it represents. [2 marks]


0

( )
The rate of change of the height (h) of a ball above horizontal ground, measured
in metres, t seconds after it has been thrown and until it hits the ground, can be
modelled by the equation
dh = 11. 4 − 9. 8t
dt
The height of the ball when t = 0 is 1. 2 m.

3a. Find an expression for the height h of the ball at time t. [6 marks]
3b. Find the value of t at which the ball hits the ground. [2 marks]

3c. Hence write down the domain of h . [1 mark]

3d. Find the range of h . [3 marks]


A theatre set designer is designing a piece of flat scenery in the shape of a hill.
The scenery is formed by a curve between two vertical edges of unequal height.
One edge is 2 metres high and the other is 1 metre high. The width of the scenery
is 6 metres.
A coordinate system is formed with the origin at the foot of the 2 metres high
edge. In this coordinate system the highest point of the cross‐section is at (2, 3. 5)
.

A set designer wishes to work out an approximate value for the area of the
scenery (A m2 ).

4a. Explain why A < 21. [1 mark]


4b. By dividing the area between the curve and the x‐axis into two [4 marks]
trapezoids of unequal width show that A > 14. 5, justifying the direction
of the inequality.

In order to obtain a more accurate measure for the area the designer decides to
model the curved edge with the polynomial
h(x)= ax3 + bx2 + cx + da, b, c, d ∈ R where h metres is the height of the
curved edge a horizontal distance x m from the origin.

4c. Write down the value of d. [1 mark]


4d. Use differentiation to show that 12a + 4b + c = 0. [2 marks]

4e. Determine two other linear equations in a, b and c. [3 marks]

( )
4f. Hence find an expression for h(x). [3 marks]

4g. Use the expression found in (f) to calculate a value for A. [2 marks]
A company’s profit per year was found to be changing at a rate of
dP = 3t2 − 8t
dt
where P is the company’s profit in thousands of dollars and t is the time since the
company was founded, measured in years.

5a. Determine whether the profit is increasing or decreasing when t = 2. [2 marks]

4
5b. One year after the company was founded, the profit was 4 thousand [4 marks]
dollars.
Find an expression for P (t), when t ≥ 0.

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