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Standard Level

14.02 Area between curves

Worksheet
Areas under multiple curves
1

For each of the following pairs of functions:

i

Graph the functions on the same set of axes.

ii

Hence, find the area of the region bounded by the functions and the x-axis.

a

x + y = 2 and y = x.

b

x + y = 6 and y = 2x.

2

For each of the following pairs of functions:

i

Graph the functions on the same set of axes.

ii

State the value(s) of x at which the two functions intersect.

iii

Hence, find the area of the region bounded by the functions and the x-axis.

a

y=2x and y=(x-4)^2.

b

y = x^{2} and y = \left(x - 2\right)^{2}

3

Find the area of the region bounded by the following pairs of functions and the x-axis:

a
y = (x-1)^{2} and y = \left(x + 3\right)^{2}
b

y=x^2 - 4 and y=-2x+11.

c

y=x^3 and y=-3x+14.

d

y=(x+2)^2 and y=-4x-3.

4

Find the area of the region bounded between the curve y = \sqrt{x + 5}, the line y = - x - 3 and the x-axis:

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

Find the area of the region bounded between the curve y = \sqrt{3-x}, the line y=\dfrac{2x+8}{3} and the x-axis:

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

Find the area of the region bounded by the curves y = e^{ 2 x} and y = e^{ - x }, the x-axis, and the lines x = - 2 and x = 2:

-3
-2
-1
1
2
3
x
1
2
3
y
7

Find the area of the region bounded by y=x^2 + 1, y=-\dfrac{x}{2}+6, the x-axis, and the y-axis.

8

The diagram shows the shaded region bounded by y = 3, y = 0, y = 6 x - x^{2} - 8, x = 0 and x = 6:

Find the area of the shaded region.

9

Consider the following graph:

a

Find the area of the shaded region B.

b

Find the area of the shaded region A.

c

What is the ratio of the areas A:B?

10

Consider the following graph:

Find k such that the two shaded regions have equal area.

11

Find the exact area between the graph of y = \sqrt{4 - x^{2}} and the lines x = 2 and y = 2.

12

Consider the functions y = e^{\frac{x}{6}} and y = e^{\frac{3}{2}}.

a

Find the x-coordinate of the point of intersection.

b

Calculate the exact area bounded by y = e^{\frac{x}{6}}, the y-axis and the line y = e^{\frac{3}{2}}.

13

Consider the curve y = e^{x}.

a

Find the area bound by the curve, the x-axis, the y-axis, and the line x = 3.

b

Find the equation of the tangent to the curve y = e^{x} at the point where x = 3.

c

Find the exact area enclosed between y = e^{x}, the x-axis, the y-axis, and the tangent at x = 3.

Areas between curves
14

Find the area enclosed between the lines \\y = 2 x, y = \dfrac{1}{3} x and x = 6:

15

The diagram shows the shaded region bounded by y = 4 - x^{2}, y = 1 - x^{2} and the x-axis:

Find the area of the shaded region.

16

Consider the graph of the functions y = x^{2} and y = x^{4}:

Find the area enclosed between the two curves.

-2
-1
1
2
x
1
2
3
4
y
17

Consider the graph of the functions \\ f\left(x\right) = x \left(x - 6\right)^{2} and g\left(x\right) = x^{2}:

a

State the values of x at which the curves intersect.

b

Hence, find the total area bounded between the curves.

c

Find the area bounded by the curves and the x-axis.

-1
1
2
3
4
5
6
7
8
9
x
40
80
y
18

For each of the following pairs of functions:

i

Graph the functions on the same set of axes.

ii

State the values of x at which the line and the curve intersect.

iii

Hence, find the area enclosed between the line and the curve.

a

y = x^{2} and y = x + 2

b

y = x \left(x - 4\right) and y = x

c

y = - x^{2} + 8 and y = - x + 2

d

y = x \left(x - 4\right)^{2} and y = x

e

y = - x \left(x - 3\right)^{2} and y = - x

19

For each of the following pairs of functions:

i

Find the values of x at which the two curves intersect.

ii

Find the area enclosed between the two curves.

a

y = x^{2} - 48 and y = - \left(x - 2\right)^{2} + 4

b

y = x and y = \left(x - 5\right)^{3} + 5

c

y = 4 x - 12 and y = x \left(x - 3\right)^{2}

d

y = 2 x and y = x^{2} - 15

20

Consider the functions y = - 2 x \left(x - 4\right) and y = - x + 4.

a

Graph the functions on the same set of axes.

b

State the values of x at which the curve and the line intersect.

c

Hence, find the area enclosed between the curves.

d

Find the small area enclosed between the curve, the line and the y-axis.

21

Consider the functions y = x^{2} and y = 8 - x^{2}.

a

Graph the functions on the same set of axes.

b

State the values of x at which the curves intersect.

c

Hence, find the area bounded between the curves.

d

Find the small area bounded between the curves and the x-axis, correct to one decimal place.

22

The diagram shows the graphs of y = \sin x and y = 3 \sin x:

a

Determine whether the following integral could be performed to calculate the area of the region bound by the curves, A.

i

\int_{0}^{\pi} \left( 3 \sin x - \sin x\right) dx

ii

\int_{0}^{\pi} 3 \sin x dx - \int_{0}^{\pi} \sin x dx

iii

\int_{0}^{\pi} 3 \sin x dx + \int_{0}^{\pi} \sin x dx

iv

2 \int_{0}^{\pi} \sin x dx

v

\int_{0}^{\pi} \left( 3 \sin x + \sin x\right) dx

vi

4 \int_{0}^{\pi} \sin x dx

b

Find the area of the shaded region A.

c

What is the ratio of the areas A:B?

23

Find the area of the region bounded by the curves y = \sin x and y = \cos x, and the lines x = 0 and x = \dfrac{\pi}{4}:

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
x
-1
1
y
24

Find the area enclosed by the two curves of \\y = \sin 2 x and y = \cos x on the interval \\ \dfrac{\pi}{6} \leq x \leq \dfrac{\pi}{2}.

\frac{1}{6}π
\frac{1}{3}π
\frac{1}{2}π
x
-1
-0.5
0.5
1
y
25

Find the area between the two curves of y = \sin 8 x and y = \cos 8 x on the interval \left[0,\dfrac{\pi}{16}\right].

\frac{1}{32}π
\frac{1}{16}π
\frac{3}{32}π
x
-1
-0.5
0.5
1
y
26

Find the area enclosed by the two curves of f\left(x\right) = 5 \cos 2 x and g\left(x\right) = 5 \sin \left(\dfrac{x}{2}\right) between x = \dfrac{\pi}{5} and x = \pi. Round your answer to two decimal places.

\frac{1}{5}π
\frac{2}{5}π
\frac{3}{5}π
\frac{4}{5}π
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
27

Find the area of the region bounded by f\left(x\right) = 4 \cos x, g\left(x\right) = - 4 \cos \left(x - \dfrac{\pi}{2}\right), the y-axis and x = \dfrac{3 \pi}{4}.

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
x
-4
-3
-2
-1
1
2
3
4
y
28

Find the area of the region bounded by the line y = 2 \pi - x and the curve y = - 2 \sin x and the x-axis between x = 0 and x = \dfrac{2 \pi}{3}.

29

Consider the functions y = \sin x and y = \cos x.

a

Graph the functions on the same set of axes in the interval \left[0, \pi\right].

b

Hence, find the area between the two curves in this interval.

30

Consider the functions y = \sin 4 x and y = \cos 4 x for 0 \leq x \leq \dfrac{\pi}{2}.

a

Find the x-coordinates of the points of intersection of the two curves in the given interval.

b

Graph the two functions on the same set of axes.

c

Hence, find the area bounded by the two curves between the points of intersection found in part (a).

31

Consider the functions f\left(x\right)=\sin 4 x and g\left(x\right)=1 - \cos 4 x.

a

Graph the functions from 0 to \dfrac{\pi}{2} on the same set of axes.

b

For what values of x is f\left(x\right) = g\left(x\right)?

c

Evaluate \int_{0}^{\frac{\pi}{2}} \left(1 - \cos 4 x - \sin 4 x\right) dx.

d

Find the area between the two curves on the interval \left[0, \dfrac{\pi}{2}\right].

32

Find the exact area between the following graphs and lines:

a

y = x^{3} - 8, x = 2, and y = - 8

b

y = e^{x}, y = e^{ - x }, and x = 3

c

y = e^{ 5 x}, y = e^{ - 5 x }, x = - 2, and x = 2

d

y = x, y = e^{1 - x}, and x = 3

33

Consider the functions f \left( x \right) = e^{x} + 12 and g \left( x \right) = 4 x + e^{3}.

a

Find the x-coordinate of the point of intersection that is to the right of origin.

b

Find the area in the first quadrant bounded by the two curves and the y-axis.

34

Consider the graphs of f \left( x \right) = x^{2} and g \left( x \right) = \sqrt{x}:

a

Find the x-coordinates of the points of intersection.

b

Find the shaded area bounded by the two curves.

c

Complete the following table for the area bounded by f \left( x \right) = x^{n} and g \left( x \right) = \sqrt[n]{x}.

n2358
\text{Area}
d
Make a conjecture for the value of the area between the curves f \left( x \right) = x^{n} and g \left( x \right) = \sqrt[n]{x} for positive integer values of n.
e
Prove your conjecture.
35

Consider the graph below which has the following properties:

  • Functions f \left( x \right) and g \left( x \right) intersect at x = 0, x = 2 and x = 3

  • Region A has an area of 22 square units

  • Regions A and B have a combined area of 56 square units

  • \\ \int_{2}^{3} f \left( x \right) dx = - 18 and \int_{2}^{3} g \left( x \right) dx = - 13

a

Find \int_{0}^{2} f \left( x \right) dx.

b

Find \int_{0}^{2} \left(f \left( x \right) - g \left( x \right) \right) dx.

c

Find the area of region C.

d

Find the area of region D.

e

Find \int_{0}^{3} \left(f \left( x \right) - g \left( x \right) \right) dx.

36

Consider the graph below which has the following properties:

  • Functions f \left( x \right) and g \left( x \right) intersect on the x-axis at points p,q, and r

  • Region A has an area of 16\text{ units}^2

  • Regions A and B have a combined area of 26\text{ units}^2

  • \int_{p}^{r} f \left( x \right) dx = 11 and \int_{p}^{r} g \left( x \right) dx = - 9
a

Find the value of \int_{p}^{q} f \left( x \right) dx.

b

Find the value of \int_{p}^{q} g \left( x \right) dx.

c

Find the area of region C.

d

Find the value of \int_{p}^{r} \left(f \left( x \right) - g \left( x \right) \right) dx.

37

Consider the graph of the functions f \left( x \right) = 3 x and g \left( x \right) = \dfrac{x^{2}}{2} together with the line x = 8:

\\

Region P is the area enclosed by f and g.

Region Q is the area enclosed by f and g and x = 8.

a

Find the area of region P.

b

Find the area of region Q.

c

f \left( x \right) is redefined such that f \left( x \right) = ax and the area of region P is half the area of region Q. Calculate the value of a that makes the statement true.

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