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.