Using a double integral, find the volume of the solid that's bounded by the cylinder z=x^2 and below by the region enclosed by the parabola y=2-x^2 and the line y=x

To find the volume of the solid bounded by the given surfaces using a double integral, we need to set up the integral for the volume.

First, let's find the limits of integration for the two variables, x and y.

Since the solid is bounded by the cylinder z = x^2, the limits of x and y are determined by the intersection of the parabola y = 2 - x^2 and the line y = x.

Setting these two equations equal to each other, we have:

2 - x^2 = x

Rearranging the equation, we get:

x^2 + x - 2 = 0

Factoring the equation, we have:

(x + 2)(x - 1) = 0

So, x = -2 or x = 1.

Since y = x, the limits of y are determined by these x values.

Next, let's set up the integral in terms of x and y for the volume of the solid.

The volume V is given by:

V = ∬(R) f(x, y) dA

where R is the region enclosed by the parabola y = 2 - x^2 and the line y = x, and f(x, y) = x^2.

To evaluate this integral, we need to switch the order of integration to make the calculation easier:

V = ∫[a, b] ∫[g(x), h(x)] x^2 dy dx

where a = -2, b = 1, and g(x) = x, h(x) = 2 - x^2.

Now, let's calculate the inner integral with respect to y:

∫[g(x), h(x)] x^2 dy = x^2 ∫[x, 2 - x^2] dy

= x^2 * (2-x^2 - x)

= x^2 (2 - 2x^2 - x)

= 2x^2 - 2x^4 - x^3

Finally, we integrate the result with respect to x:

V = ∫[a, b] (2x^2 - 2x^4 - x^3) dx

= [2/3 x^3 - 2/5 x^5 - 1/4 x^4] from -2 to 1

Calculating this expression, we get:

V = [(2/3 - 2/5 - 1/4) - (8/3 - 16/5 + 16/4)]

V = [(-1/60) - (256/60)]

V = -257/60

Therefore, the volume of the solid bounded by the given surfaces is -257/60.