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7.04 Approximate areas of integration

Worksheet
Approximation of areas
1

Use the trapezoidal rule to find an approximation for each integral:

a
\int_{2}^{5} 3x^{2} \, dx
b
\int_{0}^{2} (2x^{3}-1) \, dx
c
\int_{2}^{4} dx
2

Find an approximation to \int_{0}^{1}x^{3} \, dx using the trapezoidal rule with:

a

1 subinterval

b

2 subintervals

3

Given the table of values, find the approximate value of each definite integral:

a

\int_{0}^{8}f \left(t\right) \, dt

t02468
f\left( t \right)371193
b

\int_{0}^{40}f \left(x \right) \, dx

x010203040
f\left( x \right)350410435450460
c

\int_{-4}^{2}f\left(x\right) \,dx

x-4-3-2-1012
f\left( x \right)045310112
4

Use the trapezoidal rule with 2 subintervals to find an approximation for the following to three decimal places:

a
\int_{3}^{4}\ln\, 2x\, dx
b
\int_{0}^{6}\dfrac{dx}{x+3}
5

Find an approximation to the following integrals, rounding your answers to three decimal places:

a

\int_{2}^{3}e^{2x} \,dx using 3 subintervals

b

\int_{0}^{2} \left(4x^{2}-16 \right) \,dx using 4 subintervals

c

\int_{0}^{1}\sqrt{2x}\,dx using 5 subintervals

d

\int_{1}^{3}\dfrac{dx}{x^{3}} using 4 subintervals

e

\int_{2}^{5} \dfrac{2}{3x-4}\,dx\,using 6 subintervals

6

Use one application of the trapezoidal rule to approximate \int_{2}^{3} \dfrac{e^{x}}{x}\, dx to one decimal place.

7

Use three applications of the trapezoidal rule to approximate \int_{0}^{9} e^{ - x^{2} }\, dx to one decimal place.

8

Approximate \int_{0}^{8} 8 x \, dx by using four rectangles of equal width whose heights are the values of the function at the midpoint of each rectangle.

9

Approximate \int_{1}^{5} \dfrac{1}{x} \, dx by using four rectangles of equal width whose heights are the values of the function at the right endpoint of each rectangle.

10

In the following graph, the interval \left[0, 8\right] is partitioned into four subintervals \left[0, 2\right], \left[2, 4\right], \left[4, 6\right], and \left[6, 8\right]:

a

Approximate the area A using rectangles for each subinterval whose heights are equal to the function values of the left side of the subintervals.

b

Approximate the area A using rectangles for each subinterval whose heights are equal to the function values of the right side of the subintervals.

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9
x
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y
11

The function y = 3 \ln x has been graphed:

Use two applications of the trapezoidal rule to approximate the area bound by the curve, the x-axis and and x = 6. Round your answer to one decimal place.

1
2
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x
-6
-5
-4
-3
-2
-1
1
2
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y
12

Consider the function y = e^{x^{2}}.

a

Determine y''.

b

Using two applications of the trapezoidal rule, approximate the area bound by the curve and the x-axis between x = 0 and x = 2. Round your answer to one decimal place.

c

Is the approximation given by the trapezoidal rule an underestimate or an overestimate of the actual area? Explain your answer.

13

Consider the integral \int_{1}^{5} \left(\ln x + 4\right) \, dx.

Approximate the area bounded by the curve, the x-axis, x = 1 and x = 5 using two applications of the trapezoidal rule. Round your answer to two decimal places.

14

Use the trapezoidal rule to find the approximate area of the irregular figure below:

Applications
15

The following piece of land has straight boundaries on the east, west and south borders and is bounded by a creek to the north. The land has been divided into two sections so we can use the trapezoidal rule to approximate the area:

a

Find the approximate area of the piece of land by using two applications of the trapezoidal rule.

b

During a heavy storm, 35.2 \text{ mm} of rain fell. Find the volume of water that falls on this land. Round your answer to the nearest cubic metre.

16

A surveyor made the following diagram with measurements for a property she was mapping out. On the west side of the property is a river:

a

Find the approximate total area of the property by using three applications of the trapezoidal rule.

b

The average weekly rainfall is 34 \text{ mm}. Calculate the total volume of water that falls on the land in cubic metres. Round your answer to two decimal places.

17

A river has its depths marked out at equal intervals of 9 \text{ m}. The depths are 0, 12, 14, 17, 5, and 0 \text{ m} respectively. Find the approximate area of the cross section of the river.

18

The elevation values of a mountain are recorded at equal intervals of 250 \text{ m}. The heights are shown in the diagram:

Find the approximate area of the cross section of the mountain.

19

The diagram shows the cross-section of a river. The depths of the river are marked at 2-metre intervals:

Using three applications of the trapezoidal rule, approximate the area of the cross-section of the river to one decimal place.

20

Use four applications of the trapezoidal rule to approximate the area of the cross-section of the following river:

21

The following shape has measurements given in metres. Use the trapezoidal rule to find the area in hectares.

22

A garden is 49 \text{ m} long. At 7 \text{ m} intervals, the width of the garden was given by the following measurements:

0 \text{ m}, \, 2.9\text{ m}, \, 5.2\text{ m}, \, 6.6\text{ m}, \, 5.6\text{ m}, \, 4.3\text{ m}, \, 3\text{ m}, \, 2.5\text{ m}

Using the trapezoidal rule, approximate the area of the garden to two decimal places.

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MA12-3

applies calculus techniques to model and solve problems

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