Question:

Let curve C be described by the equation y = e^(x/2), where x ≥ 0. The area enclosed between curve C and the x-axis, from x = 0 to x = k, is rotated around the x-axis to create a solid of revolution. Find the value of k that will result in the maximum volume of the solid.

Answer:

To find the value of k that will result in the maximum volume of the solid, we need to determine the bounds of integration, set up the integral for finding the volume, and then differentiate with respect to k to maximize the volume.

First, let's find the bounds of integration. We want to find the value of k where the curve C intersects the x-axis. Since y = 0 when the curve intersects the x-axis, we can substitute y = 0 into the equation y = e^(x/2) to find the value of x.

0 = e^(x/2)

Taking natural logarithm on both sides, we get:

ln(0) = ln(e^(x/2))

0 = x/2

x = 0

So, the curve C intersects the x-axis at x = 0.

Next, we will set up the integral for finding the volume of the solid of revolution. The volume of a solid of revolution using the disk method is given by the formula:

V = π ∫[a, b] [f(x)]^2 dx

In this case, the integrand is [f(x)]^2 = [e^(x/2)]^2 = e^x.

Therefore, the volume of the solid is given by:

V = π ∫[0, k] e^x dx

Next, we integrate the function e^x with respect to x:

V = π [e^x]∣[0, k]

V = π (e^k - e^0)

V = π (e^k - 1)

To find the value of k that maximizes the volume, we differentiate the expression for V with respect to k and set it equal to zero:

dV/dk = π e^k

Setting dV/dk to zero:

π e^k = 0

e^k = 0

There is no value of k that satisfies e^k = 0, so there is no maximum volume.

Therefore, the solid of revolution has no maximum volume.