Question

What did I do wrong?





An object is formed so that its base is the quarter circle

y = sqrt(64 − x^2)

in the first quadrant, and its cross sections along the x-axis are squares. What is the volume of the object? (Assume the axes are measured in centimeters.)



I have already set up my equation as

1/64 * pi * r^2
r= 64 - x^2
limits of integration (0, 8)

For the integral, I have


pi/64 * integral from 0 to 8 of 64 - x^2 dx


I got pi/64 (64x - x^3/3)

That, evaluated at 8 gives me 34901/2083.

Answer 1

if you draw the square at x, it has a side of length 2y, or

2√(64-x^2)

So, the area of the square there is

4(64-x^2)

The volume is the sum of all those squares of thickness dx, so

v = ∫[0,8] 4(64-x^2) dx