How do I solve this using Cylindrical shells?

Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.:

y = sqrt(8 + x^2), y=0, x=0, x=1

how do I solve this problem?

Answer 1

So we want to take our function, revolve it around the y-axis, and calculate the volume. If we are using the cylindrical shell method, then our integral will sum the volumes of infinitely many "rings" or cylindrical shells. Since we are rotating about the y-axis, the radius of each shell is just x, and the shell's height is f(x) = sqrt(8+x²). Because the shell is infinitely thin, its volume is essentially the circumference of the circle formed multiplied by the height. That is, V = 2πxf(x).

We thus take the integral from 0 to 1 of 2πx*sqrt(8+x²)dx. We can then use substitution to more easily solve this integral. We can set u = 8+x² which gives du = 2xdx. We may notice immediately that du is already present in our integral in terms of x, allowing easy substitution. Our limits of integration would give x = 0 -> u = 8, x + 1 -> u = 9. Plugging in, we can now take the integral from 8 to 9 of π*sqrt(u)du = (2π/3)*u^2/3 from 8 to 9. This gives (18 - 32*sqrt(2)/3)π which is approximately 9.158.