Compute the volume of the solid formed by revolving the regi

Compute the volume of the solid formed by revolving the region bounded by y x y 4-x about (a) the x-axis; (b) y 4.

Solution

we find the points of intersection
x²=4-x²

x²=2

x=+/-2

So x is betweeonm 0 and 2

and on this interval 4-x² >=x²

It is easier to use cylindrical shells

The volume is

2_0..2 x((4-x²) -x²) dx

2(2x²-x⁴/2)

=2(4-2)=4

For part b, the distance to the axis x=4 is 4-x, so the integral is

2_0..2 (4-x)((4-x²) -x²) dx=

...

=2(322/3-2)