Question

A solid is formed by rotating the region bounded by the curve y=e^(-3x/2) and the x-axis between x=0 and x=1, around the x-axis. Assuming the solid has constant density, find its mass.

Answer 1

To find the mass of the solid formed by rotating the given curve around the x-axis, we can use the formula for the volume of a solid of revolution. First, we need to find the coordinates where the curve intersects the x-axis. Then, we can calculate the volume using the definite integral, and finally find the mass using the volume and the constant density.

Step-by-step explanation:

To find the mass of the solid formed by rotating the given curve around the x-axis, we can use the formula for the volume of a solid of revolution:

V = π ∫[a, b] (f(x))^2 dx

First, we need to find the coordinates where the curve intersects the x-axis. Setting y = 0, we have:

e^(-3x/2) = 0

Since e is a positive constant, this equation has no real solutions. Therefore, the solid formed by rotating the curve does not intersect the x-axis and has no empty space. This means the entire region under the curve is rotated to form the solid.

Next, we can calculate the volume:

V = π ∫[0, 1] (e^(-3x/2))^2 dx

This integral can be solved by using substitution or integration by parts. Evaluating the definite integral will give us the volume of the solid. Finally, since the solid has constant density, we can use the volume to find the mass using the equation:

m = ρV