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5.01 Area under a curve

Worksheet
Approximate areas under graphs
1

The following graph shows the function y= e^{x}:

a

Find an estimate for the area between the function and the x-axis for \\0 \leq x \leq 4, giving your answer to three decimal places and using:

i

The left-endpoint approximation with 4 rectangles.

ii

The right-endpoint approximation with 4 rectangles.

iii

Technology and the left-endpoint approximation with 50 rectangles.

b

Use integration to find the exact area between the function and the x-axis for 0 \leq x \leq 4.

c

Calculate the percentage error of the approximation in (a) part (iii) compared to the exact area. Round your answer to the nearest percent.

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2

The following graph shows the function y = \sqrt{x + 1}:

a

Find an estimate for the area between the function and the x-axis for \\2 \leq x \leq 6, giving your answer to three decimal places and using:

i

The left-endpoint approximation with 4 rectangles.

ii

The right-endpoint approximation with 4 rectangles.

iii

Technology and the right-endpoint approximation with 40 rectangles.

b

Use integration to find the area between the function and the x-axis for \\ 2 \leq x \leq 6. Round your answer to four decimal places.

c

Calculate the percentage error of the approximation in (a) part (iii) compared to area calculated by integration in part (b). Round your answer to two decimal places.

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3

The following graph shows the function f \left( x \right) = \cos x:

a

Find an estimate for the area between the function and the x-axis for \\0 \leq x \leq \dfrac{\pi}{2}, giving your answer to three decimal places and using left-endpoint approximation with:

i

3 rectangles.

ii

5 rectangles.

iii

Technology and 50 rectangles.

b

Use integration to find the exact area between the function and the x-axis for 0 \leq x \leq \dfrac{\pi}{2}.

c

Calculate the percentage error of the approximation in (a) part (iii) compared compared to the exact area. Round your answer to the nearest percent.

\frac{1}{4}π
\frac{1}{2}π
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4

Consider the function f\left(x\right) = xe^{-x} and the given graph of y=f\left(x\right):

a

Find f'\left(x\right).

b

Hence, show that: \\ \int xe^{-x}=-xe^{-x}-e^{-x}+c.

c

Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 5, giving your answer to three decimal places and using:

i

2 rectangles of equal width.

ii

4 rectangles of equal width.

d

Find the exact area bounded by f\left(x\right) and the x-axis over 1 \leq x \leq 5, using part (b).

e

Calculate the percentage error for the two estimates given in part (c). Round your answer to two decimal places.

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5

Consider the function f\left(x\right) =e^{x}+10-6e^{-x} and the given graph of y=f\left(x\right):

a

Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 3 using 4 rectangles of equal width. Give your answer to four decimal places.

b

Calculate an underestimate for the area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 3 using 4 rectangles of equal width. Give your answer to four decimal places.

c

Given f''\left(x\right)>0 for x>0.9, which estimate will be closer to the exact area bounded by f\left(x\right) and the x-axis for \\1 \leq x \leq 3. Explain your answer.

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6

Consider the function f\left(x\right) =\dfrac{5x}{\sqrt{x^2+1}} and the given graph of y=f\left(x\right):

a

Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\0 \leq x \leq 2 using 2 rectangles of equal width. Give your answer to three decimal places.

b

Calculate an overestimate for the area bounded by f\left(x\right) and the x-axis for \\2 \leq x \leq 4 using 2 rectangles of equal width. Give your answer to three decimal places.

c

Which of the estimates calculated in (a) and (b) will be closer to the exact area being estimated? Explain your answer.

d

Would an underestimate or overestimate using 4 rectangles of equal width provide a closer estimate to the exact area bounded by f\left(x\right) and the \\x-axis for 0 \leq x \leq 4. Explain your answer.

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Definite integration
7

The following graph shows the function y = x^{2} - 4:

a

Find the area A.

b

Find the area B.

c

Find the ratio of the areas A:B.

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8

The following graph shows the function y = 3x^{2}:

Let A define the area under the curve over the interval \left[0, k \right] and B define the area under the curve over the interval \left[k, 2k \right]. Find the ratio of the areas A:B.

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The following graph shows the function y = \cos x:

Find k such that ratio A:B is 1:1.

10

Consider the function f \left( x \right) = e^{x}. Let A define the area under the curve over the interval \left[0, k \right] and B define the area under the curve over the interval \left[0, 2k \right]. Find the relationship between A and B.

11

The curve y = 8 \pi - 2 x + \sin x is shown passing through the point P \left( \pi, 6 \pi \right). A straight line joins the origin to point P.

a

Find the exact value of \int_{0}^{\pi} \left( 8 \pi - 2 x + \sin x\right) dx.

b

Find the exact area of A.

c

Find the exact area of B.

d

Determine the ratio A:B in the form \\1:k. Round k to two decimal places.

12

Consider the function f \left( x \right) = 6 a x - 6 x^{2}, for a \geq 0.

a

For a = 1, sketch the function.

b

Find the area bound by the curve and the x-axis.

c

Use your calculator to complete the table of the area bound by the graph and the x-axis for different values of a.

a123510
\text{Area }(\text{units})^2
d
Make a conjecture for the value of the area bounded by f \left( x \right) = 6 a x - 6 x^{2} and the \\x-axis, for a \geq 0.
e
Prove your conjecture.
13

Consider the function f \left( x \right) = k x^{n} \left(1 - x\right) on the interval \left[ 0 ,1 \right], where n is a positive integer.

a

For n = 1, find the value of k if the area under the curve on the interval \left[0, 1 \right] is 1\text{ unit}^2.

b

Using technology, find the area under the curve y = x^{2} \left(1 - x\right) on the interval \left[ 0, 1 \right].

c

Hence, find the value of k that would create an area of 1\text{ unit}^2 under the curve of f \left( x \right) over the interval \left[ 0, 1 \right], when n = 2.

d

Complete the following table showing the value of k required for different values of n, if the area on the interval \left[0, 1\right] is 1\text{ unit}^2:

n12345
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e
Make a conjecture for the value of k in terms of n, required if the area under f\left(x\right) on the interval \left[0, 1\right] is 1\text{ unit}^2.
f
Prove your conjecture.
14

Consider the curve y = f_k \left( x \right) where f_k \left( x \right) = \cos \left( kx \right) and k is a positive integer.

The domain of the function f_k \left( x \right) is dependent on k which is 0 \leq x \leq \dfrac{\pi}{2 k}.

a

The given graph is of f_1 \left( x \right), where \\k = 1, thus f_1 \left( x \right) = \cos \left( 1 x \right) and the domain is 0 \leq x \leq \dfrac{\pi}{2}.

Find the area between y = f_1 \left( x \right) and the x-axis over the domain of the function.

b

Consider y = f_k \left( x \right) for k = 2:

i

State the function forf_2 \left( x \right).

ii

State the domain of f_2 \left( x \right).

iii

Find the area between y = f_2 \left( x \right) and the x-axis over the domain of the function.

c

Find the area under f_3 \left(x\right) over its domain.

d
Make a conjecture for the value of the area bounded by y = f_k \left(x\right) and the \\x-axis over 0 \leq x \leq \dfrac{\pi}{2 k}, for any positive integer value of k.
e
Prove your conjecture.
\frac{1}{4}π
\frac{1}{2}π
x
y
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