MAT01B1:Areas between curves (and some volumes)

Dr Craig

5 September 2018

My details:

I acraig@uj.ac.za

I Consulting hours:

Monday 14h40 – 15h25

Thursday 11h20 – 12h55

Friday 11h20 – 12h55

I Office C-Ring 508

https://andrewcraigmaths.wordpress.com/

(Or, just google ‘Andrew Craig maths’.)

Some new curves:

x = y2

x = y2 − 9

When sketching curves such as these, pay

attention to the sign (+ve or −ve) of x and

y and also the direction of the shifts. Plug in

x = 0 and y = 0 to get reference points.

The area between two curves

Height of rectangle centered at xi is given by:

(y-value of top curve) − (y-value bottom curve)

or, in this case:f (xi)− g(xi)

Example: find the area bounded above by

the curve y = ex, bounded below by y = x

and bounded on the left by x = 0 and on the

right by x = 1.

Example: find the area bounded by the

curves y = x2 and y = 2x− x2.

Area between curves

The area between two curves y = f (x)

and y = g(x), and between x = a and

x = b is

A =

∫ b

a

|f (x)− g(x)| dx

Example 5

Find the area of the region bounded by

y = sinx, y = cosx, x = 0 and x = π/2.

Solution: 2√2− 2.

Example 6

Find the area enclosed by the line y = x− 1

and the parabola y2 = 2x + 6.

Solution:∫ 4

−2

[(y + 1)−

(y2

2− 3

)]dy = 18

Using two methods: find the area between

the curves y2 = 4− x and 4y = −x + 4.

Integrating with respect to x we get:∫ 4

−12

(√4− x + x

4− 1)dx

Integrating with respect to y we get:∫ 4

0

(4y − y2) dy

In both cases the answer is32

3.

Using both methods: find the area of the

region between x + y2 = 4 and x− y = 2.

Solution: the problem on the previous slide

is much easier to solve if you integrate with

respect to y.

Either method will give you A = 412.

Volumes

Familiar volume calculations

We can calculate the volume of many shapes

by multiplying the area of the base by the

height of the shape. However, this only works

if the shape at the base is extended upwards

at right angles and remains constant.

When calculating the area of a disk at x, we

will use the notation A(x).

We will sometimes integrate with respect to

y and then we will write A(y) for the area of

a disk at y.

Calculating the volume of a sphere:

Calculating the volume of a sphere: