Question

Find the volume of the solid that lies between the graph of y=

1−x 2​ and y=− 1−x 2​ whose cross sections perpendicular to the x-axis are squares.

a) Sketch a slice on the region to the right.

b) Draw the cross section and label the slice you drew in a)

c) Write the equation for the cross section in terms of the slice.

d) Write the integral expression for the volume of the solid

Answer 1

To determine the volume of the solid, calculate the integral of the square of the difference between the two parabolic functions over the interval where they intersect.

Step-by-step explanation:

The student is asking to find the volume of a solid whose cross-sections perpendicular to the x-axis are squares, and it lies between the graph of y = 1 - x2 and y = -1 - x2. For such a figure the side length of each square cross-section would be determined by the difference in the y-values of the two curves.

Steps to Find the Volume:

The integral expression for the volume V of the solid is:
V = ∫-11 (2 - 2x2)2 dx.