Question

On the same coordinate axes, sketch the graph of y = eˣ and y = 3 + 4e^(-x). If the region bounded by the curves and the y-axis is revolved through 2π radians about the x-axis, find the volume of the solid generated.

a. π²
b. 4π²
c. 2π²
d. 8π²

Answer 1

To find the volume of the solid generated by revolving the region bounded by the curves y = e^x and y = 3 + 4e^(-x) about the x-axis, we can use the method of cylindrical shells. The volume element of a cylindrical shell is given by V = 2πrhΔx, where r is the perpendicular distance from the x-axis to the shell, h is the height of the shell, and Δx is the thickness of the shell.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the curves y = e^x and y = 3 + 4e^(-x) about the x-axis, we can use the method of cylindrical shells. The volume element of a cylindrical shell is given by V = 2πrhΔx, where r is the perpendicular distance from the x-axis to the shell, h is the height of the shell, and Δx is the thickness of the shell.

First, let's find the points of intersection of the two curves. Set y = e^x equal to y = 3 + 4e^(-x) and solve for x:

e^x = 3 + 4e^(-x)

Now, substitute these values of x into the expressions for r and h:

r = y = e^x

h = y - (3 + 4e^(-x))

Finally, integrate the volume element over the interval where the region is bounded:

V = ∫(2πrhΔx)