Find the exact volume of a solid

Question

Find the exact volume of a solid whose base is bounded by the curves y=e^x, y=0, x=-1 and x-1 and that has rectangular cross-sections perpendicular to the x-axis, using rectangles of twice the height of their bases.

Answer 1

V=∫^b_aA(x)dx

A(x) = base×height = e^x(2e^x)

V=2∫¹_-1 e^2xdx

Which is easy enough to integrate.

Answer 2

Cross section area A(x) = 2e^2x∫; volume v = ∫_-1¹2e^2xdx = e^2x_-1¹ = e² - e^-2

Answer 3

Hi Madison,

If we draw this bounded region on a sheet of paper, the rectangles that form the solid would appear to be coming out of the paper toward you. We're going to find the area of each rectangle, multiply them by their thickness to form a volume, and sum up all the volumes to find the overall volume of the solid.

The area of a rectangle is:

A = B * H

Since the rectangles are perpendicular to the axis, the lengths of their bases will be bounded by the distance from the x-axis to the function e^x. The base length, therefore, is:

B = e^x

The height is said to be "twice the length of the base", so

H = 2e^x

A = e^x * 2e^x = 2e^2x

Since the squares are perpendicular to the x-axis, the thickness of each square will be dx. So, the volume of each individual square is

V = 2e^2xdx

This is the volume of only one slice of the solid. We want to find the volume of the entire solid from x = -1 to x = 1. So we'll integrate the function from -1 to 1.

2 ∫_-1¹ e^2x dx

= e^2x |_-1¹

= e² - e^-2

Hope this helps!