AREAS INVOLVING X AND Y AXES WORKSHEET
1.
VOLUMES OF REVOLUTION
Find the area of the shaded regions below:
1
AREAS INVOLVING X AND Y AXES WORKSHEET
2.
VOLUMES OF REVOLUTION
A section of the curve 𝑦 = 3√4 − 𝑥 is drawn below.
Find the area of the region bounded by the curve and the coordinate axes by considering the
region between the curve and the:
a) 𝑥 −axis
b) 𝑦 −axis
3.
a) Find the area of the region enclosed by 𝑦 = √𝑥, the 𝑥 −axis and the line 𝑥 = 16.
b) Find the area of the region enclosed by 𝑦 = √𝑥, the y−axis and the line 𝑥 = 4.
c) Find the sum of the results in (a) and (b), and explain what this area represents.
4.
The diagram below shows 𝑦 = √4 − 𝑥 and a region bounded by the lines 𝑥 = 4 and 𝑦 = 2.
Find the area of the region.
5.
The diagram below shows a sketch of 𝑦 = 8𝑥 − 2𝑥 ! .
Find the area of the shaded region.
2
AREAS INVOLVING X AND Y AXES WORKSHEET
6.
VOLUMES OF REVOLUTION
Bob is asked to find the area of the region bound by the curve 𝑦 = 3√4 + 𝑥 and the
coordinate axes.
He produces the diagram above, follows the usual steps and obtains the integral
"
𝑦!
2 3 − 45 𝑑𝑦
9
#
Explain why this integral does not give the required area.
7.
Find the area enclosed by the curve 𝑦 = 𝑥 $ , the 𝑦 −axis and:
a) The lines 𝑦 = 1 and 𝑦 = 8
b) The lines 𝑦 = −8 and 𝑦 = −1
c) The lines 𝑦 = −8 and 𝑦 = 8
8.
Find the area of the region enclosed by the following curves and the 𝑦 −axis.
a) 𝑥 = 4𝑦 − 𝑦 !
b) 𝑥 = 𝑦 ! − 𝑦
c) 𝑥 = −𝑦 ! + 8𝑦 − 12
d) 𝑥 = −𝑦 ! + 7𝑦 − 10
9.
Find the area of the following regions below:
a) 𝑦 = ln(3 − 𝑥) and both axes
b) 𝑦 = cos %& 𝑥 and both axes
3
AREAS INVOLVING X AND Y AXES WORKSHEET
10.
VOLUMES OF REVOLUTION
Complete the steps given to find the area of the shaded region below.
a)
b)
c)
d)
Write down an integral representing the area of the shaded region.
Write down the integral representing the area of B and hence find the area.
Find the area of the rectangle shown in dotted lines.
Hence, find the area of |𝐴|.
11.
Find the area of the following regions:
12.
By drawing a sketch where necessary and shading the appropriate region, evaluate the
following integrals:
&
a) ∫# sin%& 𝑥 𝑑𝑥
&
b) ∫# cos %& 𝑥 𝑑𝑥
&
c) ∫# tan%& 𝑥 𝑑𝑥
4