Find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y=3/10x² , y=13/10-x²

Answer 1

I'll choose to integrate in the x-direction using the "washer" method. Imagine a series of washers with inside radius r_i and outside radius r_o that are "dx" thick.

The intersections of the functions are the limits of integration so to find them, set the functions equal to each other:

3/10x² = 13/10 - x²

0 = 13/10 - x² - 3/10x²

0 = 13/10 - 13/10x²

0 = 13/10(1 - x²)

0 = 13/10(1 + x)(1 - x)

x = -1 and x = 1

r_o = 13/10 - x²

r_i = 3/10x²

V = _-1∫¹[π(r_o² - r_i²)dx

V = π• _-1∫¹[r_o² - r_i²]dx

V = π• _-1∫¹[(13/10 - x²)² - (3/10x²)²]dx

V = π• _-1∫¹[169/100 - 13/5x² + x⁴ - 9/100x⁴]dx

V = π• _-1∫¹[169/100 - 13/5x² + 91/100x⁴]dx

V = π• (169/100x - 13/15x³ + 91/500x⁵) evaluated between -1 and 1

V = π•[(169/100 - 13/15 + 91/500) - (-169/100 + 13/15 - 91/500)]

V = π•[169/100 - 13/15 + 91/500 + 169/100 - 13/15 + 91/500]

V = π•[338/100 - 26/15 + 182/500]

V = π•[5070/1500 - 2600/1500 + 546/1500]

V = π•[3016/1500)

V = 754/375π ≈ 6.3167