A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 6

Finding areas

112 marks

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Date

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Quick questions to start.

A1
Needed today
Find the exact value of $\int_{0}^{1} e^{2 x} d x$ .
[2 marks]
Answer
A2
Needed today
Find the exact value of $\int_{0}^{π /4} sec^{2} x d x$ .
[2 marks]
Answer
A3
Common slip
Find $\int_{- 1}^{1} x^{3} d x$ . Then find the area between $y = x^{3}$ , the $x$ -axis and the lines $x = - 1$ and $x = 1$ .
[2 marks]
Answer
A4
Needed today
Use integration by parts to find the exact value of $\int_{0}^{1} x e^{x} d x$ .
[3 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 area of the region bounded by the curve $y = x e^{- x}$ , the $x$ -axis and the line $x = 2$ .

Hint The curve meets the

x

-axis at

x = 0

, so the limits are

0

and

2

. Use integration by parts with

u = x

B1
Find the exact area of the region bounded by the curve $y = x sin x$ and the $x$ -axis for $0 \leq x \leq π$ .
[4 marks]
Answer

Example 2

Find the total area of the shaded regions between the curve $y = cos 2 x$ and the $x$ -axis for $0 \leq x \leq \frac{π}{2}$ .

Hint The curve crosses the axis at

x = \frac{π}{4}

. Integrate the two parts separately and add the sizes.

B2
The curve $y = e^{x} - 2$ crosses the $x$ -axis at $A$ . Find the $x$ -coordinate of $A$ , and find the exact total area between the curve and the $x$ -axis for $0 \leq x \leq 2$ .
[5 marks]
Answer

Example 3

Find the exact area of the region enclosed by $y = sin x$ and $y = cos x$ between their first two points of intersection for $x > 0$ .

Hint Solve

sin x = cos x

to get the limits. Then integrate (top curve

-

bottom curve).

B3
The curves $y = e^{x}$ and $y = 4 - 3 e^{- x}$ meet at two points. Find the $x$ -coordinates of the two points and the exact area of the region between the curves.
[6 marks]
Answer

Example 4

A curve has parametric equations $x = 3 t^{2}$ , $y = 2 t$ , $t \geq 0$ . Find the area of the region bounded by the curve, the $x$ -axis and the line $x = 12$ .

Hint Area

= \int y \frac{d x}{d t} d t

. Change the limits:

x = 0

gives

t = 0

and

x = 12

gives

t = 2

B4
A curve has parametric equations $x = sin t$ , $y = sin 2 t$ , $0 \leq t \leq \frac{π}{2}$ . Find the exact area of the region between the curve and the $x$ -axis.
[5 marks]
Answer

Practice

A-Level

C1
Leo is asked for the area between $y = x (x - 3)$ and the $x$ -axis for $0 \leq x \leq 4$ . He writes: " $\int_{0}^{4} (x^{2} - 3 x) d x = \frac{64}{3} - 24 = - \frac{8}{3}$ , so the area is $\frac{8}{3}$ ." Explain his mistake and find the correct area.
[5 marks]
Answer

C2
A curve has parametric equations $x = 1 + t^{2}$ , $y = 2 t$ , $t \geq 0$ . Sam finds the area between the curve, the $x$ -axis and the lines $x = 1$ and $x = 5$ by writing $\int_{1}^{5} 2 t \times 2 t d t = \frac{496}{3}$ . Explain his mistake and find the correct area.
[4 marks]
Answer
C3
The region $R$ is bounded by the curve $y = ln x$ , the $y$ -axis, the $x$ -axis and the line $y = 2$ . Find the exact area of $R$ .
[4 marks]
Answer
C4
Find the exact area of the region bounded by the curve $y = \frac{2}{( x + 1 ) ( x + 3 )}$ , the coordinate axes and the line $x = 3$ . Give your answer in the form $ln k$ .
[5 marks]
Answer
C5
Use the substitution $u = 2 x + 1$ to find the exact area of the region bounded by $y = \frac{x}{2 x + 1}$ , the $x$ -axis and the line $x = 4$ .
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The diagram shows part of the curve $y = x^{2} ln x$ , $x > 0$ . The curve meets the $x$ -axis at the point $A$ . The region $R$ is bounded by the curve, the $x$ -axis and the line $x = e$ .
(a)
Find the coordinates of $A$ .
[2 marks]
Answer
(b)
Find $\int x^{2} ln x d x$ .
[4 marks]
Answer
(c)
Hence find the exact area of $R$ , giving your answer in the form $\frac{a e ^{3} + b}{9}$ .
[3 marks]
Answer
Total for D1: 9 marks

D2
Priya is asked for the area of the region between the curve $y = x^{3} - 4 x$ and the $x$ -axis. She writes: "The curve crosses the axis at $x = - 2$ and $x = 2$ , and $\int_{- 2}^{2} (x^{3} - 4 x) d x = 0$ , so the area is $0$ ."
Priya is wrong.
(a)
Explain the two mistakes in her working.
[2 marks]
(b)
Find the area of the region between the curve and the $x$ -axis.
[4 marks]
Answer
(c)
The line $y = 5 x$ meets the curve at the origin and at a point $P$ with $x > 0$ . Find the area of the region bounded by the line and the curve between $O$ and $P$ .
[4 marks]
Answer
Total for D2: 10 marks
D3
The curve $C$ has parametric equations $x = t - sin t$ , $y = 1 - cos t$ , $0 \leq t \leq 2 π$ . The region $S$ is bounded by $C$ and the $x$ -axis. Show that the area of $S$ is $3 π$ .
[8 marks]
D4
The diagram shows the curves $y = 2 sin x$ and $y = tan x$ for $0 \leq x < \frac{π}{2}$ . The shaded region lies between the curves.
(a)
Show that the curves meet where $x = 0$ and where $x = \frac{π}{3}$ .
[3 marks]
(b)
Show that $\int tan x d x = - ln ∣ cos x ∣ + c$ .
[2 marks]
(c)
Hence find the exact area of the shaded region, in the form $a + b ln 2$ .
[4 marks]
Answer
Total for D4: 9 marks
D5
$f (x) = \frac{x + 7}{( x + 1 ) ( x + 3 )}$ , $x \geq 0$ .
(a)
Express $f (x)$ in partial fractions.
[3 marks]
Answer
(b)
Hence find the exact area of the region bounded by $y = f (x)$ , the coordinate axes and the line $x = 3$ , giving your answer in the form $k ln 2$ .
[4 marks]
Answer
Total for D5: 7 marks
D6
The curve $y = \frac{6 x}{x ^{2} + 1}$ and the line $y = 3 x$ enclose a region $T$ for $x \geq 0$ .
(a)
Show that the curve and the line meet where $x = 0$ and where $x = 1$ .
[2 marks]
(b)
Find the exact area of $T$ .
[5 marks]
Answer
Total for D6: 7 marks

Extension

Stretch

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

E1
For $x \geq 0$ the curve $y = e^{- x} sin x$ crosses the $x$ -axis at $x = 0, π, 2 π, \dots$ . Let $A_{n}$ be the area of the region between the curve and the $x$ -axis for $(n - 1) π \leq x \leq nπ$ . Show that $A_{n + 1} = e^{- π} A_{n}$ , and find the exact total area of all these regions.
[10 marks]
Hint 1 · What to try
Use integration by parts twice on $\int e^{- x} sin x d x$ , then rearrange to find it.
Hint 2 · The first line
At $x = nπ$ the sine is $0$ and the cosine is $(- 1)^{n}$ . Work out $A_{n}$ in general, taking the size of each integral.
Hint 3 · The full method
The areas form a geometric series. Add them all with the sum to infinity.

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the area between a curve and the x-axis, splitting the region where the curve crosses the axis.
I can find the area between two curves, or a curve and a line, from their points of intersection.
I can find the area under a parametric curve using the integral of y times dx/dt, with limits in t.
I can choose substitution, parts or partial fractions to do the integration an area needs.

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