Question

Find the volume of the solid obtained by rotating the region bounded by the curve y = e⁽⁻ˣ²⁾, y = 0, x = 0, x = 1 about the y-axis.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = e^(-x^2), y = 0, x = 0, x = 1 about the y-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = e^(-x^2), y = 0, x = 0, x = 1 about the y-axis, we can use the method of cylindrical shells.

The volume of a cylindrical shell is given by the formula V = 2πrhΔx, where r is the distance from the y-axis to the curve, h is the height of the shell, and Δx is the thickness of the shell.

Integrating from x = 0 to x = 1, we can set up the integral as follows:

V = ∫[0,1] 2πx e^(-x^2) dx

Simplifying and evaluating the integral gives us the volume of the solid.