Find the exact surface area of revolution about the x-axis. Given y=sinpix, 0<x<1

Question

The interval that was given is a closed interval. So, it should be 0 is less than greater than x is less than greater than 1.

Answer 1

Use the formula:

Where r(x) represents the radius as you move from the lower bound (0) to the upper bound (1), in this case that's just sin(πx). And where f(x) = sin(πx) meaning f '(x) = πcos(πx) so f '(x)² = π²cos²(πx).

Plug all that in and use a calculator to solve the integral and you get 7.779.

If you want to do it manually it would involve u substitutions but I didn't complete the integration so I'm not 100% sure how it would go.