First thing to do is calculate your limits of integration. You can do that by finding where your curves meet.

In the second equation, you can see that y = -3 so substitute y = -3 into the first equation and solve for x.

x^2 + 3*-3 = 0

x = (+/-)3, so we're integrating from x=-3 to 3.

Since our limits of integration are in terms of x, we will have to use a method that uses an integral with respect to x.

Rotating about the y axis & integrating with respect x means we'll be using the shell method.

Now we know what our integral will look like: 2(pi)*INTEGRAL(r(x)h(x)dx) (from a to b)

where r(x) is the distance from the axis of rotation and h(x) is the height at x.

Because our axis is the y-axis, distance from it is simply 'x.'

For h(x) we have to ask our self which curve is on top, and we observe that x^2 - 9 < -3 on the interval [-3, 3]. so our height function will be

h(x) = (-3) - (x^2 - 9).

Now we know all the parts needed to calculate volume:

V = 2(pi)*INTEGRAL(x*[(-3) - (x^2 - 9)]dx) from -3 to 3.