Topics
11. Integration
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Standard level

11.02 Evaluate areas using geometry

Lesson
Worksheet
Practice
Worksheet
Geometric areas
1

Calculate geometrically, the area bounded by the following functions and the x-axis over the given domain:

a

2 \leq x \leq 5

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b

0 \leq x \leq 7

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c

0 \leq x \leq 4

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d

0 \leq x \leq 9

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2

Calculate geometrically, the exact value of the following definite integrals:

a

\int_{0}^{6} \left(6 - x\right) dx

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b

\int_{0}^{12} f \left( x \right) dx

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f(x)
c

\int_{0}^{16} f \left( x \right) dx

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f(x)
d

\int_{1}^{8} \left(10 - x\right) dx

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e

\int_{3}^{5} \left( 2 x + 1\right) dx

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f

\int_{0}^{16} f \left( x \right) dx

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f(x)
g

\int_{ - 6 }^{6} \sqrt{36 - x^{2}} dx

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-5
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h

\int_{0}^{6} \sqrt{36 - x^{2}} dx

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i

\int_{0}^{16} f \left( x \right) dx

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f(x)
j

\int_{0}^{12} \sqrt{36 - \left(x - 6\right)^{2}} dx

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3

For each of the following functions:

i

Sketch a graph of f \left( x \right).

ii

Hence, calculate geometrically the area bounded by the curve and the x-axis over the given domain:

a

f \left( x \right) = 6 - 6 x for 0 \leq x \leq 1

b

f \left( x \right) = 3 x + 4 for 0 \leq x \leq 3

c

f(x)=\begin{cases} x & \text{for } 0\leq x \lt 4\\ 8-x & \text{for } 4\leq x\leq 8 \end{cases}

d

f(x)=\begin{cases} 2x & \text{for } 0\leq x \leq 3\\ 6 & \text{for } 3\lt x\lt 6\\ 18-2x & \text{for } 6\leq x \leq 9 \end{cases}

Approximate areas
4

Consider the function f \left( x \right) = x^{2} - 3 x + 10.

For each of the following intervals, state whether the right endpoint approximation method will underestimate or overestimate the area under the curve. Explain your answer.

a

\left[ 3, 8 \right]

b

\left[ -6, -2 \right]

5

Consider the function f \left( x \right) = x^{2} + 5.

For each of the following intervals, state whether the left endpoint approximation method will underestimate or overestimate the area under the curve. Explain your answer.

a

\left[ 2, 6 \right]

b

\left[ -5, -1 \right]

6

Consider the function f \left( x \right) = 10 e^{x}.

State the approximation method that will give an underestimate of the true area on any given interval. Explain your answer.

7

Consider the function f \left( x \right) = -3 x^{2} - 24 x + 1.

a

State whether the right endpoint approximation to the area will overestimate or underestimate on the following intervals.

i

\left[ -10, -5 \right]

ii

\left[ -4, 2 \right]

iii

\left[ 0, 2 \right]

iv

\left[ -6, -4 \right]

b

State whether the left endpoint approximation to the area will overestimate or underestimate on the following intervals.

i

\left[ -10, -5 \right]

ii

\left[ -4, 2 \right]

iii

\left[ 0, 2 \right]

iv

\left[ -6, -4 \right]

8

The function f \left( x \right) = 5 x is defined on the interval \left[0, 6\right].

a

Graph f \left( x \right).

b

Find the area under f \left( x \right) by partitioning \left[0, 6\right] into 3 sub-intervals of equal length using:

i

Left endpoint approximation

ii

Right endpoint approximation

c

Find the area under f \left( x \right) by partitioning \left[0, 6\right] into 6 sub-intervals of equal length using:

i

Left endpoint approximation

ii

Right endpoint approximation

d

Find the actual area under the curve on the interval \left[0, 6\right].

9

The function f \left( x \right) = - 4 x + 12 is defined on the interval \left[0, 3\right].

a

Graph f \left( x \right).

b

Find the area under f \left( x \right) by partitioning \left[0, 3\right] into 3 sub-intervals of equal length using:

i

Left endpoint approximation

ii

Right endpoint approximation

c

Find the area under f \left( x \right) by partitioning \left[0, 3\right] into 6 sub-intervals of equal length using:

i

Left endpoint approximation

ii

Right endpoint approximation

d

Find the actual area under the curve on the interval \left[0, 3\right].

10

The interval \left[0, 8\right] is partitioned into 4 sub-intervals \left[0, 2\right], \left[2, 4\right],\left[4, 6\right], and \left[6, 8\right].

Find the area under the curve on the interval \left[0, 8\right] using:

a

Left endpoint approximation

b

Right endpoint approximation

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11

Use left endpoint approximation to find \int_{1}^{9} 2 x^{2} dx by using 4 rectangles of equal width.

12

Use right endpoint approximation to find \int_{1}^{5} \dfrac{1}{x} dx by using 4 rectangles of equal width.

Give your answer as a simplified fraction.

13

Approximate \int_{3}^{15} \left( 4 x + 6\right) dx by using 4 rectangles of equal width and using the method:

a

Left endpoint approximation

b

Right endpoint approximation

Use technology
14

Use technology to find the exact value of the following definite integrals:

a
\int_{2}^{5} 2x + 3 \ dx
b
\int_{0}^{3} e^x \ dx
c
\int_{-3}^{3} -x^2 + 9 \ dx
15

Consider the function f \left( x \right) = 0.5 x^{2}.

a

Complete the table to estimate the area between the function and the x-axis for \\1 \leq x \leq 3 using the left endpoint approximation method. Round your answers to three decimal places.

n510100100010\,000
A_L \text{ units}^2
b

Use technology to evaluate \int_{1}^{3} 0.5x^2 \ dx and hence confirm that the exact area is the limit of A_L as n gets larger.

16

Consider the function f \left( x \right) = \sqrt{4 - x^{2}}.

a

Complete the table to estimate the area between the function and the x-axis for \\0 \leq x \leq 2 using the right endpoint approximation method. Round your answers to five decimal places.

n5101001000
A_R \text{ units}^2
b

Use technology to evaluate \int_{0}^{2} \sqrt{4 - x^{2}} \ dx and hence confirm that the exact area is the limit of A_R as n gets larger.

17

Consider the function f \left( x \right) = - x^{2} + 3 x.

a

Sketch a graph of f(x).

b

State the x-values that define the region bounded by the curve and the x-axis. Write your answer as an inequality.

c

Complete the table to estimate the area of the region bounded by the function and the \\x-axis using the right endpoint approximation method. Round your answers to four decimal places.

n510100100010\,000
A_R \text{ units}^2
d

Use technology to confirm that the exact area is the limit of A_R as n gets larger.

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