Question

How do you use substitution to integrate `2piy(8-y^2/3)dy`?

Answer 1

You can simply multiply it out. No complicated substitution required. Looks like the Shell Method?

`V = 2piintxf(x)dx`

Switch out `x` for `y`. Looks like `f(y) = 8 - y^2/3`.

`= 2piint_a^b 8y - y^3/3dy`

`= 2pi[4y^2 - y^4/12]|_(a)^(b)`

`= color(blue)(2pi[(4b^2 - b^4/12) - (4a^2 - a^4/12)])`