Question

Find the volume of the solid under the plane z = 3x + y and above the region determined by y= x^7 and y=x

Answer 1

To find the volume of the solid under the plane z = 3x + y and above the region determined by y = x^7 and y = x, set up a double integral over the region and evaluate it.

Step-by-step explanation:

To find the volume of the solid under the plane z = 3x + y and above the region determined by y = x^7 and y = x, we need to calculate the double integral over the given region.

1. First, find the intersection points of the two curves y = x^7 and y = x.
2. Set up the limits of integration for x and y based on the intersection points.
3. Then, determine the limits of integration for z, which are the minimum and maximum values of the plane z = 3x + y over the region.
4. Set up the double integral for the volume using the appropriate limits.
5. Finally, evaluate the double integral to find the volume.

For example, the intersection points of y = x^7 and y = x are at x = -1, 0, and 1. The limits of integration for x would be -1 to 1, and the limits for y would be x^7 to x. The limits for z would be the minimum and maximum values of z = 3x + y over the region.