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7.07 Compound areas and the area between two curves

Worksheet
Areas under multiple curves
1

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

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2

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

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3

Consider the functions x + y = 2 and y = x.

a

Graph the functions on the same number plane.

b

Hence, find the area bounded between the two lines and the x-axis.

4

Consider the functions x + y = 6 and y = 2x.

a

Graph the functions on the same number plane.

b

Hence, find the area bounded between the two lines and the x-axis.

5

Consider the functions y = x^{2} and y = \left(x - 2\right)^{2}.

a

Graph the functions on the same number plane.

b

State the value of x at which the curves intersect.

c

Hence, find the area enclosed between the curves and the x-axis.

6

Consider the functions y = (x-1)^{2} and y = \left(x + 3\right)^{2}.

a

Graph the functions on the same number plane.

b

State the value of x at which the curves intersect.

c

Hence, find the area enclosed between the curves and the x-axis.

7

Find the area of the region bounded by the following pairs of functions and the x-axis, to two decimal places:

a

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

8

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

Areas between curves
9

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.

10

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.

11

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

12

For each of the following:

i

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

ii

Find the area between the two curves.

a

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

b

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

c

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

13

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

a

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

b

Find the area enclosed between the two curves.

14

For each of the following:

i

Sketch the functions on the same Cartesian plane.

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

15

The following diagram shows the curves y = - x^{2} + 4 x - 4 and y = x^{2} - 8 x + 12 meeting at the points \left(2, 0\right) and \left(4, - 4 \right):

Find the area of the shaded region.

16

The following diagram shows the curves y = x^{2} and y = x^{4}:

Find the area of the shaded region.

17

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

a

Graph the functions on the same number plane.

b

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

c

Hence, find the area enclosed between the curves, correct to one decimal place.

d

Find the small area enclosed between the curve, the line and the y-axis. Round your answer to one decimal place.

18

Consider

a

Graph the functions y = x^{2} and y = 8 - x^{2} on the same Cartesian plane.

b

State the values of x at which the curves intersect.

c

Hence, find the area bounded between the curves, correct to one decimal place.

d

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

19

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

20

Find the area between the graph of y = x^{3} - 8 and the lines x = 2 and y = - 8.

21

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

a

State the values of x at which the curves intersect.

b

Hence, find the total area bounded between the curves. Round your answer to one decimal place.

c

Find the area bounded by the curves and the x-axis. Round your answer to one decimal place.

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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-7

applies the concepts and techniques of indefinite and definite integrals in the solution of problems

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