11.8 Finding areas: Areas under parametric curves

Example

Exercise

Extra content

The curve C has parametric equations

x = t(1 + t), y = 1 ____ 1 + t , t > 0

Find the exact area of the region R, bounded by C, the x-axis and the lines x = 0 and x = 2.

Area = ∫y dx = ∫  y dx ___ dt dt

x = t(1 + t), so dx ___ dt = 1 + 2t

When x = 0, t(1 + t) = 0, so t = 0 or t = −1

When x = 2, t(1 + t) = 2

t2 + t − 2 = 0

(t + 2)(t − 1) = 0, so t = −2 or t = 1

So Area = ∫ 0 1 y dx ___

dt dt = ∫ 0 1

1 ____ 1 + t (1 + 2t) dt

= ∫ 0 1 (2 − 1 ____

1 + t ) dt

= [2t − ln | 1 + t|] 0 1

= (2 − ln 2) − (0 − ln 1)= 2 − ln 2

You can use integration to find the area under a curve defined by parametric equations. It is often easier to integrate with respect to the parameter.

R

2O

y

x

C

Use a change of variable to write the integral in terms of the parameter, t. Using the chain rule,

you can replace dx with dx ___ dt

dt .

x = t + t2

Write 1 + 2t _____ 1 + t in the form A + B ____

1 + t

Dividing 2t + 1 by t + 1 gives 2 with remainder −1.

You will be integrating with respect to t so you need to convert the limits from values of x to values of t. Use the parametric equation for x, and choose solutions that are within the domain of the parameter, t > 0.

Watch out

1 The curve C has parametric equations x = t3, y = t2, t > 0. Show that the exact area of the region bounded by the curve, the x-axis and the lines x = 0 and x = 4 is k 3 √

_ 2 , where k is a

rational constant to be found.

2 The curve C has parametric equations x = sin t, y = sin 2t, 0 > t > π __ 2

The finite region R is bounded by the curve and the x-axis. Find the exact area of R. (6 marks)

3 This graph shows part of the curve C with parametric equations x = (t + 1)2, y = 1 _ 2 t3 + 3, t > −1

P is the point on the curve where t = 2. The line S is the normal to C at P.a Find an equation of S. (5 marks)The shaded region R is bounded by C, S, the x-axis and the line with equation x = 1.b Using integration, find the area of R. (5 marks)

P

E/P

O

y

x

C

R

E/P

xO 1

R

PCS

y

4 The diagram shows the curve C with parametric equations x = 3t2, y = sin 2t, t > 0.

a Write down the value of t at the point A where the curve crosses the x-axis. (1 mark)

b Find, in terms of π , the exact area of the shaded region bounded by C and the x-axis. (6 marks)

5 The curve shown has parametric equationsx = 5 cos θ, y = 4 sin θ, 0 < θ < 2π

a Find the gradient of the curve at the point P at which θ = π __ 4 (3 marks)b Find an equation of the tangent to the curve

at the point P. (3 marks)c Find the exact area of the shaded region bounded by the tangent PR, the curve and the x-axis.

(6 marks)

6 The curve C has parametric equations x = 1 − t2, y = 2t − t3, t ∈ ℝ

The line L is a normal to the curve at the point P where the curve intersects the positive y-axis. Find the exact area of the region R bounded by the curve C, the line L and the x-axis, as shown on the diagram. (7 marks)

7 The curve shown in the diagram has parametric equationsx = t − 2 sin t, y = 1 − 2 cos t, 0 < t < 2π

a Show that the curve crosses the x-axis where t = π __ 3 and t = 5π ___ 3 (3 marks)

The finite region R is enclosed by the curve and the x-axis, as shown shaded in the diagram.

b Show that the area R is given by ∫ π __ 3 5π ___ 3

(1 − 2 cos t) 2 dt (3 marks)

c Use this integral to find the exact value of the shaded area. (4 marks)

E/P

O

R

y

C

Ax

E/P

xO

P

R

y

E/P

R

O x

y

L

CP

E/P

R

O x

y

ANSWERS

1 Area = ∫ y dx ___ dt

dt = ∫ 0 3 √ _

4 t 2 (3 t 2 ) dt = 3 __ 5 ( 3 √ _

4 ) 5 = 3 __ 5 2 10

__ 3

= 3 __ 5 ( 2 3 ) ( 2 1 __ 3 ) = 24 __ 5 3 √ _

2

2 2 __ 3

3 a x + y = 16 b 61.85

4 a π __ 2 b 3π ___

2

5 a − 4 __ 5 b y − 2 √ _

2 = − 4 __ 5 (x − 5 ___ √ _

2 )

c 10 − 5π ___ 2

6 41 __ 60

7 a 2cost = 1 ⇒ cost = 1 __ 2 ⇒ t = π __ 3 or t = 5π ___

3

b ∫ π __ 3

5π ___ 3

y dx ___

dt dt = ∫ π __

3 5π ___ 3

(1 − 2 cos t) (1 − 2 cos t) dt

= ∫ π __ 3

5π ___ 3

(1 − 2 cos t) 2 dt

c 4π + 3 √ _

3