Topic 3.2: Integrals

Check the sign of the function before you call an integral an area: split at the roots, and take upper minus lower between two curves.

A lesson for this topic

Four-option practice

Question 1Practice

Which of the following is

\int (6x^2 - 4x + 5)\,dx

Question 2Practice

Which of the following is

\int \frac{3}{x^2}\,dx

Question 3Practice

Which of the following is

\int 4\sqrt{x}\,dx

Question 4Practice

Which of the following is

\int e^{4x}\,dx

Question 5Practice

Which of the following is

\int \sin(3x)\,dx

Question 6Practice

Which of the following is

\int (2x - 5)^4\,dx

Question 7Practice

Which of the following is

\int \left(4\cos(2x) - 6e^{-3x}\right)dx

Question 8Practice

Which of the following is

\int (x + 2)(3x - 1)\,dx

Question 9Practice

Which of the following is

\int \frac{x^3 - 4}{x^2}\,dx

Question 10Practice

The curves

y

= f(x)

with

f'(x)

= 3x^2 - 2

form a family. Which of the following functions belongs to the same family as

g(x)

= x^3 - 2x + 7

, that is, has the same derivative?

Question 11Practice

A function satisfies

f'(x)

= 6x^2 - 2x

and

f(1)

= 5

. Which of the following is

f(x)

Question 12Practice

A function satisfies

f''(x)

= 12x

f'(1)

= 4

and

f(1)

= 3

. What is the value of

f(2)

Question 13Practice

Given that

\frac{d}{dx}(x\sin x)

= \sin x + x\cos x

, which of the following is

\int x\cos x\,dx

Question 14Practice

Given that

\frac{d}{dx}\left((x^2 + 1)^3\right) = 6x(x^2 + 1)^2

, which of the following is

\int x(x^2 + 1)^2\,dx

Question 15Practice

A function satisfies

f'(x)

= 3e^{0.2x}

and

f(0)

= 2

. What is the value of

f(10)

, correct to two decimal places?

Question 16Practice

A table of values of f: x = 0, 1, 2, 3, 4 gives f(x) = 2, 3, 6, 11, 19.
$x$	0	1	2	3	4
$f(x)$	2	3	6	11	19

The table gives values of a function

f

that is increasing for

0

\le x

\le 4

f(0)

= 2

f(1)

= 3

f(2)

= 6

f(3)

= 11

and

f(4)

= 19

. Four rectangles of width 1 are drawn, each with height equal to the value of

f

at the left end of its interval. What estimate of

\int_0^4 f(x)\,dx

do they give?

Question 17Practice

The graph shows

y

= 16 - x^2

for

0

\le x

\le 4

. The shaded area is estimated with four rectangles of width 1, each with height equal to the value of the function at the right end of its interval. Which gives the estimate and compares it correctly with the exact area?

Question 18Practice

Which definite integral is equal to

\lim_{n \to \infty} \sum_{i=1}^{n} \left(1 + \frac{2i}{n}\right)^3 \frac{2}{n}

Question 19Practice

The graph of

y

= f(x)

for

0

\le x

\le 7

is made of straight line segments joining

(0, 0)

(1, 4)

(2, 0)

(3, -2)

(4, -2)

(5, 0)

(6, 3)

and

(7, 0)

. What is the value of

\int_0^7 f(x)\,dx

Question 20Practice

Given

\int_1^4 f(x)\,dx

= 7

and

\int_1^6 f(x)\,dx

= 3

, what is the value of

\int_4^6 \left(2f(x) + 1\right)dx

Question 21Practice

Let

F(x)

= \int_2^x (t^3 - 2t)\,dt

. What is the value of

F'(3)

Question 22Practice

Which of the following is

\frac{d}{dx}\int_x^5 \cos(t^2)\,dt

Question 23Practice

What is the value of

\int_1^2 (6x^2 - 4x + 1)\,dx

Question 24Practice

What is the exact value of

\int_0^{\pi/4} \left(4\cos(2x) + \sin x\right)dx

Question 25Practice

What is the exact value of

\int_0^1 (e^{2x} + 1)\,dx

Question 26Practice

The graph shows

y

= 3 + 2x - x^2

. What is the area of the shaded region, between the curve and the

x

-axis from

x

= 0

x

= 3

Question 27Practice

What is the total area of the regions enclosed by the curve

y

= x^2 - 6x + 8

, the

x

-axis and the lines

x

= 1

and

x

= 5

Question 28Practice

The graph shows

y

= x^2

and

y

= 2x + 3

, which meet at

(-1, 1)

and

(3, 9)

. What is the area of the shaded region enclosed between them?

Question 29Practice

The graph shows

y

= x^3 - 2x^2

and

y

= 3x

, which meet where

x

= -1

x

= 0

and

x

= 3

. What is the total area of the two shaded regions enclosed between them, correct to two decimal places?

Question 30Practice

A particle moves in a straight line with velocity

v(t)

= 3t^2 - 12t + 9

metres per second, for

t

\ge 0

seconds. What is its displacement over the first 4 seconds?

Question 31Practice

A particle moves in a straight line with velocity

v(t)

= t^2 - 6t + 8

metres per second, for

t

\ge 0

seconds. What is the total distance it travels during the first 5 seconds?

Question 32Practice

Water flows into a tank on a wheatbelt farm at a rate of

20 - 2t

litres per minute, where

t

is the time in minutes after the tap is opened, for

0

\le t

\le 10

. How much water flows into the tank in the first 6 minutes?

Question 33Practice

The marginal revenue from selling handmade surfboards is

R'(x)

= 50e^{-0.01x}

dollars per board, where

x

is the number of boards sold. What is the total revenue from selling the first 100 boards, correct to the nearest cent?

Question 34Practice

A particle moves in a straight line with acceleration

a(t)

= 6t - 4

metres per second squared. At

t

= 0

its velocity is 3 metres per second and its position is

-2

metres. What is its position at

t

= 2

seconds?

Question 35Practice

A cyclist moves along a straight path. The velocity-time graph is made of straight line segments joining

(0, 0)

(2, 6)

(5, 6)

(7, 0)

and

(9, -4)

, with

t

in seconds and

v

in metres per second. At

t

= 0

the cyclist is 5 m from the start of the path, in the positive direction. How far from the start, in the positive direction, is the cyclist at

t

= 9

Question 36Practice

A particle moves in a straight line with velocity

v(t)

= 4\cos(0.5t)

metres per second, for

0

\le t

\le 5

seconds. What total distance does it travel in these 5 seconds, correct to two decimal places?

Question 37Practice

The area between the curve

y

= e^{0.05x}

, the

x

-axis and the lines

x

= 0

and

x

= k

is 30 square units, where

k

> 0

. What is the value of

k

, correct to two decimal places?

Short response questions

A curve from its gradient functionShort response · 9 marks

A curve

y

= f(x)

has gradient function

f'(x)

= 3x^2 - 6x - 9

and passes through the point

(2, -10)

(a) Determine

\int (3x^2 - 6x - 9)\,dx

. Include the constant

+ c

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Determine

f(x)

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

x

-coordinates of the stationary points of the curve, separated by a comma. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Determine the

y

-coordinate of the stationary point where

x

= 3

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) A second curve,

y

= g(x)

, has the same gradient function,

g'(x)

= 3x^2 - 6x - 9

, and passes through the point

(2, 0)

. Describe how the graph of

y

= g(x)

is related to the graph of

y

= f(x)

, and explain why. [2]

0 words · a sentence or two

Signed area and the area functionShort response · 10 marks

The graph shows

y

= f(x)

, where

f(x)

= x^2 - 4x + 3

, for

0

\le x

\le 4

. The curve crosses the

x

-axis at

x

= 1

and

x

= 3

(a) Determine the exact value of

\int_0^4 f(x)\,dx

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Determine the total area of the regions enclosed by the curve, the

x

-axis and the lines

x

= 0

and

x

= 4

. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

0 words · a sentence or two

(d) Let

F(x)

= \int_0^x f(t)\,dt

. State the value of

F'(2)

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Determine the value of

x

, with

0

< x

< 4

, at which

F

has a local maximum. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

The region between two parabolasShort response · 9 marks

The graph shows the curves

y

= 4 - x^2

and

y

= x^2 - 2x

, and the region enclosed between them is shaded.

(a) Determine the

x

-coordinates of the points where the curves meet, separated by a comma. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Determine the area of the shaded region. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

0 words · a sentence or two

(d) The

x

-axis divides the shaded region into two parts. Determine the area of the part below the

x

-axis, which lies between the curve

y

= x^2 - 2x

and the

x

-axis. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

Estimating an area with rectanglesShort response · 9 marks

The graph shows

y

= f(x)

, where

f(x)

= 9 - x^2

, for

0

\le x

\le 3

. The shaded area under the curve is to be estimated with six rectangles, each of width 0.5.

(a) Complete the table of values of $f(x)$ by finding the two missing values. [1]

$x$	0	0.5	1	1.5	2	2.5	3
$f(x)$	9	8.75	8		5		0

Type one value in each empty cell.

(b) Determine the estimate of the area given by the six rectangles whose heights are the values of

f

at the left end of each strip. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

f

at the right end of each strip. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Determine the exact area under the curve for

0

\le x

\le 3

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Explain why the left-end estimate is larger than the exact area, and describe how the rectangle estimates could be brought closer to the exact area. [3]

0 words · two or three sentences

A particle that turns backShort response · 11 marks

A particle moves in a straight line. Its velocity is

v(t)

= 6e^{-0.5t} - 2

metres per second at time

t

seconds, for

t

\ge 0

, and at

t

= 0

it is 3 metres to the right of a fixed point O.

(a) Determine the acceleration of the particle at

t

= 0

, in metres per second squared. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Determine the time at which the particle is momentarily at rest, correct to three decimal places. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Determine the position of the particle relative to O at

t

= 4

, in metres to the right of O, correct to three decimal places. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Determine the total distance travelled by the particle during the first 4 seconds, correct to three decimal places. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(f) Explain why the answers to (c) and (e) are different. [2]

0 words · a sentence or two

The cost of making chairsShort response · 9 marks

A workshop in Midland makes timber chairs. When

x

chairs have been made, the marginal cost is

C'(x)

= 30 + 20e^{-0.02x}

dollars per chair. The workshop's fixed costs are $500, so

C(0)

= 500

, where

C(x)

is the total cost in dollars of making

x

chairs.

(a) Determine

\int_{50}^{150} C'(x)\,dx

, correct to the nearest dollar. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Interpret the value of

\int_{50}^{150} C'(x)\,dx

in the context of the workshop. [2]

0 words · a sentence or two

C(x)

. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) The workshop has a budget of $5000. Determine the greatest number of chairs it can make for a total cost of no more than $5000. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).