The volume of the solid obtained by rotating the region enclosed by y=x^2, x=y^2

Question

about the line x=−7 can be computed using the method of discs or washers via an integral.

V = integral from a to b of ?

with limits of integration a = 0 and b = 1

Answer 1

Draw a sketch!

Using the washer method, the "washers are stacked in the y-direction so you would integrate from y = 0 to y = 1.

The outside radius of each washer is 7 + the x-value of the function y = x². Since y = x², then x = √y so the outside radius is 7 + √y. That means the area of the outside circle is A = πr² = π(7 + √y)²

The inside radius of each washer is 7 + the x-value of the function x = y² so it is 7 + y². That means the area of the outside circle is A = πr² = π(7 + y²)²

So the integral is ₀∫¹ π[(7 + √y)² - π(7 + y²)² dy which you can plug in a calculator to get the answer to. My calculator say the answer is 15.603 cubic units.

The shell method would integrate in the x-direction and would consist of a series of shells where the volume of each shell is 2πrh • dx.