Integration

P1^

6

The area between a curve and the y axis

So  far you have    calculated  areas   between curves  and the x   axis.   You can also    use 

integration to  calculate   the area    between a   curve   and the y   axis.   In  such    cases,  the 

integral    involves    dy  and not dx. It  is  therefore   necessary   to  write   x   in  terms   of  y

wherever    it  appears.    The integration is  then    said    to  be  carried out with respect to y

instead of  x.

EXAMPLE 6.14  Find  the area    between the curve   y = x − 1   and the y   axis    between y = 0   and y = 4.

SOLUTION

Instead of  strips  of  width   δx  and height  y,  you now sum strips  of  width   δy  and 

length  x   (see    figure  6.21).  

You write

A x y

y

s

=

→

∑

δ 0

lim δ

overall

rectangle

= (^) ∫ 04 x dy
= (^) ∫
4
0 (y^ + 1)dy

y^2 y
0
4
2 +









= (^) 12 square units.
y = x – 1
x
y
A
O
4
–1
y = x – 1
x
y
δy
x
O
4
–1
Figure 6.21
To integrate x with
respect to y, write x
in terms of y. For this
graph y = x – 1
so x = y + 1.