Use the shell method to find the volume of the solid obtained by rotating the region bounded by the given curves

Question

$$y=\sqrt{x}$$ $$y=0$$ $$x=1$$ about x=-1

Does this set up look alright:

$$V = \int_0^1 2 \pi (x+1)(\sqrt{x}) dx$$

Answer 1

Your setup is correct.

Note the formula for computing the volume via the shell method about the y-axis is:

$$\int_{a}^{b}2\pi x(f(x)-g(x))dx$$

You've correctly identified the circumference of the circle:

$$2\pi r= 2\pi (x+1)$$

Furthmore, since

$$f(x)=\sqrt{x}$$

and

$$g(x)=0$$

The following equation (which you listed with the correct limits of integration) is correct:

$$V = \int_0^1 2 \pi (x+1)(\sqrt{x}) dx$$