Find the volume of the solid bounded by the planes x = 1, y = 0, z = 0, theparabolic cylinder y = x2, and the surface z = xey. Sketch the region of integration

To find the volume of the solid bounded by the given planes and surfaces, we can use triple integration.

Step 1: Identify the limits of integration The limits of x are given by the plane x = 1, so x goes from 0 to 1. The limits of y are given by the parabolic cylinder y = x^2, so y goes from 0 to x^2. The limits of z are given by the surface z = xey and the plane z = 0, so z goes from 0 to xey.

Step 2: Set up the triple integral The volume V of the solid is given by the triple integral of dz dy dx over the region defined by the limits of integration. So,

V = ∫ from 0 to 1 ∫ from 0 to x^2 ∫ from 0 to xey dz dy dx

Step 3: Evaluate the innermost integral The innermost integral ∫ from 0 to xey dz is simply xey - 0 = xey.

Step 4: Substitute this into the integral and evaluate the next integral V = ∫ from 0 to 1 ∫ from 0 to x^2 xey dy dx This integral can be evaluated by using the method of substitution. Let u = x^2, then du = 2x dx. The limits of integration for u are 0 and x^2. So,

V = ∫ from 0 to 1 1/2 ∫ from 0 to u uey du dx

Step 5: Evaluate the innermost integral The innermost integral ∫ from 0 to u uey du is uey - ∫ey du = uey - eyu from 0 to u = uey - eyu - 0 = 0.

Step 6: Substitute this into the integral and evaluate the final integral V = ∫ from 0 to 1 0 dx = 0

So, the volume of the solid is 0.

As for the sketch of the region of integration, it's a bit difficult to describe in text. But you can imagine it as a region in the first octant (where x, y, and z are all positive) bounded by the plane x = 1, the x-axis (y = 0), the z-axis (z = 0), the parabolic cylinder y = x^2 (which opens towards the positive y-axis), and the surface z = xey (which is a kind of warped plane that gets higher as x and y increase).