A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 13: Integration · Lesson 3

Areas under the x-axis and between curves and lines

70 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Solve $x^{2} - 3 x = x + 5$ .
[2 marks]
Answer $x = \dots\dots$
A2
Last lesson
Find the exact value of $\int_{1}^{3} (x^{2} - 1) d x$ .
[2 marks]
Answer
A3
From chapter 7
Factorise $x^{3} - 2 x^{2} - 5 x + 6$ completely.
[2 marks]
Answer
A4
From chapter 5
Find the area of the triangle between $y = 12 - 3 x$ and the axes.
[2 marks]
Answer

A-Level

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

Example 1

Find the area of the finite region bounded by the curve $y = x^{2} - 4 x$ and the $x$ -axis.

Hint Between its roots the curve is below the axis, so the integral is negative.

B1
Find the area of the finite region bounded by the curve $y = 3 x^{2} - 12$ and the $x$ -axis.
[2 marks]
Answer

Example 2

The sketch shows the curve $y = 2 x (x - 2) (x + 1)$ . Find the total area of the two shaded regions.

Hint Integrate from

- 1

0

and from

0

2

separately. One part is below the axis, so take its size.

B2
The sketch shows the curve $y = x (x + 2) (x - 4)$ . Find the total area of the two shaded regions.
[4 marks]
Answer

Example 3

The sketch shows the curve $y = x (5 - x)$ and the line $y = x + 3$ . Find the area of the shaded region between them.

Hint Find where they meet. Between those points the curve is on top, so integrate (curve minus line).

B3
The sketch shows the curve $y = 8 + 2 x - x^{2}$ and the line $y = x + 2$ . Find the area of the shaded region between them.
[4 marks]
Answer

A-Level

C1
Find the area of the finite region enclosed by the curves $y = x^{2} + 1$ and $y = 9 - x^{2}$ .
[4 marks]
Answer

C2
The finite region $R$ is bounded by the curve $y = x^{3} + 2$ , the line $y = 29$ and the $y$ -axis. Find the area of $R$ .
[3 marks]
Answer
C3
Find the value of $\int_{0}^{3} (x^{2} - 4) d x$ . Find also the total area of the regions bounded by $y = x^{2} - 4$ , the $x$ -axis and the lines $x = 0$ and $x = 3$ .
[4 marks]
Answer

Edexcel exam style

D1
Priya writes: the area of the region between $y = x^{3} - 4 x$ and the $x$ -axis is $\int_{- 2}^{2} (x^{3} - 4 x) d x = [\frac{x ^{4}}{4} - 2 x^{2}]_{- 2}^{2} = 0$
Priya is wrong. Explain her mistake, and find the correct area. Hence find the exact value of $a$ , where $a > 2$ , for which $\int_{0}^{a} (x^{3} - 4 x) d x = 0$ .
[5 marks]
Answer

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find an area below the x-axis, and an area that crosses the axis.
I can find the area between a curve and a line, or between two curves.
I can combine an integral with a triangle to find an area.

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