Question

Sketch the region bounded by the curves y = x^2, y = x^4.
1) Find the area of the region enclosed by the two curves;
2) Find the volume of the solid obtained by rotating the above region about the x-axis;
3) Find the volume of the solid obtained by rotating the above region about the horizontal line with
equation y = 1 .

Answer 1

The region bounded by the two curves is between x = -1 and x = +1. Plot the two curves and you will see why.

1) Integrate (x^2 - x^4)dx from x = -1 to x = 1

2) Integrate pi*y1^2 - pi*y2^2 dx
= pi*(x^4 - x^8)dx from x = -1 to x = 1.
y1(x) = x^2
y2(x) = x^4

3) Integrate pi[(1 - y2)^2 - (1 - y1)^2]
dx from -1 to +1