Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

y=x^2, x=y^2 about the axis x=–3

To find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 and x = y^2 about the axis x = -3, we can use the method of cylindrical shells.

First, let's visualize the curves and the axis of rotation:

1. The curve y = x^2 is a parabola that opens upwards.
2. The curve x = y^2 is a parabola that opens to the right.
3. The axis of rotation x = -3 is a vertical line passing through the point (-3, 0).

To find the volume, we'll break it down into infinitesimally thin cylindrical shells, each with a height and radius.

1. Take a small vertical strip of thickness Δx at a distance x from the y-axis.
2. The corresponding height of the shell is given by the difference between the two curves: h = x^2 - y^2.
3. The radius of the shell is the distance from the axis of rotation to the curve x = y^2, which is R = x - (-3) = x + 3.

Now we can calculate the volume of each shell using the formula for the volume of a cylindrical shell:

ΔV = 2πrhΔx

Since we're dealing with an infinitesimally thin shell, we can express the volume as a definite integral:

V = ∫(2π(x + 3)(x^2 - y^2)) dx

To evaluate this integral, we need to express y in terms of x. Since y = x^2, we substitute this expression into the integral:

V = ∫(2π(x + 3)(x^2 - x^4)) dx

Simplifying and integrating, we get the volume of the solid:

V = ∫(2π(x^3 - x^5 + 3x^2 - 3x^4)) dx

Evaluate this integral within the limits of the region of interest (where the curves intersect), and you will find the volume of the solid.