Question

Find the volume of the solid obtained by rotating the region bounded by the curve f(x) = e^(3x/8), the x-axis, the y-axis, and the line x = 1 around the y-axis?

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curve f(x) = e^(3x/8), the x-axis, the y-axis, and the line x = 1 around the y-axis, you can use the method of cylindrical shells. First, express the curve in terms of y and find the height and width of the rectangle formed by the region. Then, use the formula for the volume of a cylindrical shell to calculate the volume of the solid.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curve f(x) = e^(3x/8), the x-axis, the y-axis, and the line x = 1 around the y-axis, we can use the method of cylindrical shells. First, we need to express the curve in terms of y by solving for x: x = 8ln(y)/3. The region bounded by the curve, the x-axis, the y-axis, and the line x = 1 forms a rectangle. We need to find the height and the width of this rectangle:

1. The height is given by the difference between the maximum and minimum values of y, which are 1 and e^(3/8) respectively. So, the height is e^(3/8) - 1.
2. The width is given by the difference between the x-coordinate of the line x = 1 and the x-coordinate of the curve f(x). Since x = 8ln(y)/3, when y = 1, x = 8ln(1)/3 = 0. So, the width is 1 - 0 = 1.

Now that we have the height and width of the rectangle, we can find the volume of the solid by using the formula for the volume of a cylindrical shell: V = 2πrhΔx, where r is the radius of the shell, h is the height of the shell, and Δx is the width of the shell. In this case, the radius of the shell is the distance from the y-axis to the curve f(x), which is x = 8ln(y)/3. So, the radius is 8ln(y)/3. Therefore, the volume of the solid is:

V = 2π ∫1^(e^(3/8)) (8ln(y)/3)(e^(3/8) - 1) dy

After evaluating this integral, you will get the volume of the solid.