A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 13: Integration · Lesson 2

Definite integrals and areas under curves

70 marks

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Date

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Quick questions to start.

A1
Needed today
Solve $15 + 2 x - x^{2} = 0$ .
[2 marks]
Answer $x = \dots\dots$
A2
Last lesson
Find $\int (6 x^{2} - \frac{4}{x ^{3}}) d x$ .
[2 marks]
Answer
A3
From chapter 12
Find the stationary points of $y = x^{3} - 12 x$ .
[2 marks]
Answer
A4
From chapter 1
Find the value of $9^{\frac{3}{2}} - 4^{\frac{3}{2}}$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the exact value of $\int_{1}^{4} (3 x - \frac{2}{x ^{2}}) d x$ .

Hint Integrate, then work out (value at

4

) minus (value at

1

B1
Find the exact value of $\int_{1}^{9} \frac{x - 1}{x} d x$ .
[2 marks]
Answer

Example 2

Given that $A$ is a constant and $\int_{1}^{2} (3 x^{2} - A x) d x = A^{2}$ , show that there are two possible values of $A$ , and find them.

Hint Treat

A

as a number. Integrate and use the limits to get an expression in

A

, then set it equal to

A^{2}

B2
Given that $k$ is a constant and $\int_{0}^{3} (x^{2} - k x) d x = k^{2}$ , find the two possible values of $k$ .
[4 marks]
Answer $k = \dots\dots$

Example 3

The sketch shows the curve $y = 5 + 4 x - x^{2}$ . Find the area of the shaded region, which is bounded by the curve and the $x$ -axis.

Hint The region starts and ends where the curve meets the

x

-axis. Find those

x

values first: they are the limits.

B3
The sketch shows $y = 2 x^{2} (3 - x)$ . Find the area of the shaded region, bounded by the curve and the $x$ -axis.
[4 marks]
Answer

Practice

A-Level

C1
Given that $\int_{1}^{k} (3 x^{2} - 2) d x = 22$ , where $k > 1$ , find the value of $k$ .
[3 marks]
Answer $k = \dots\dots$

C2
Find $\int_{2}^{8} (2 - \frac{3}{x}) d x$ , giving your answer in the form $a + b 2$ , where $a$ and $b$ are integers.
[4 marks]
Answer
C3
Find the exact value of $\int_{- 2}^{- 1} (3 x^{2} + \frac{4}{x ^{3}}) d x$ .
[3 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Sam writes: $\int_{- 1}^{2} (x^{2} + 2 x) d x = [\frac{x ^{3}}{3} + x^{2}]_{- 1}^{2} = \frac{8}{3} + 4 = \frac{20}{3}$
Sam is wrong. Explain his mistake, and find the correct value of the integral. Hence find the two values of the constant $k$ for which $\int_{- 1}^{2} (x^{2} + 2 x + k) d x = 3 k^{2}$ .
[5 marks]
Answer

D2
The sketch shows part of the curve $C$ with equation $y = 9 x - x^{3}$ . The curve meets the positive $x$ -axis at $A$ . The region $R$ , shaded, is bounded by $C$ and the positive $x$ -axis.
(a)
Find the $x$ -coordinate of $A$ , and the exact coordinates of the maximum point of $C$ in the region $x > 0$ .
[5 marks]
Answer
(b)
Use calculus to find the area of $R$ .
[4 marks]
Answer
(c)
The line $x = 3$ divides $R$ into two parts. Hence show that the areas of the left part and the right part are in the ratio $5 : 4$ .
[3 marks]
Total for D2: 12 marks
D3
$f (x) = \frac{( 2 x - 3 ) ^{2}}{x}, x > 0$ . The sketch shows part of the curve $y = f (x)$ .
(a)
Show that $f (x) = 4 x^{\frac{1}{2}} - 12 + 9 x^{- \frac{1}{2}}$ .
[2 marks]
(b)
Find $\int f (x) d x$ , giving each term in its simplest form.
[3 marks]
Answer
(c)
Hence find the area of the shaded region, which is bounded by the curve, the $x$ -axis and the lines $x = 3$ and $x = 12$ . Give your answer in the form $a 3 + b$ , where $a$ and $b$ are integers, and explain why the definite integral gives the area.
[4 marks]
Answer
Total for D3: 9 marks
D4
A cyclist starts from rest. Her speed, $v$ metres per second, $t$ seconds after she starts, is modelled by
$v = k t (6 - t), 0 \leq t \leq 6$
where $k$ is a constant. The distance she travels in the $6$ seconds is $72$ metres. The distance travelled is the area under the speed-time graph.
(a)
Show that $k = 2$ . Hence find her greatest speed.
[5 marks]
(b)
Find the distance she travels while her speed is at least $10$ metres per second.
[5 marks]
Answer
Total for D4: 10 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$f (x) = x^{2} + b x + c$ , where $b$ and $c$ are constants. Given that
$\int_{0}^{1} f (x) d x = 0 and \int_{0}^{2} f (x) d x = 0,$
find $\int_{0}^{3} f (x) d x$ .
[6 marks]
Hint 1 · What to try
Each condition is an equation in $b$ and $c$ . Two equations, two unknowns.
Hint 2 · The first line
$\frac{1}{3} + \frac{b}{2} + c = 0$ and $\frac{8}{3} + 2 b + 2 c = 0$ .
Hint 3 · The full method
Halve the second: $\frac{4}{3} + b + c = 0$ . Subtract the first: $1 + \frac{b}{2} = 0$ , so $b = - 2$ and $c = \frac{2}{3}$ . Then $\int_{0}^{3} f (x) d x = 9 - 9 + 2$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can evaluate a definite integral exactly, including surd forms.
I can find an unknown constant or limit from a definite integral.
I can find the area between a curve and the x-axis.

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