Find the volume of the solid obtained by revolving the region in the first quadrant bounded by the curves y= sqrt x e^x and y= e^x about the x-axis. Draw the graphs that you can illustrate to use them

Answer 1

The graphs intersect at x = 1, so the bounds of integration are 0 to 1. The sqrtx function will be closer to the axis of rotatin, so that will provide the smaller radius, with the exponential function providing the larger. This is most easily done with respect to x, using washers, and the formula V = π ∫_a^b (r_big² - r_small²)

V = π ∫₀¹ e^2x - xe^2x dx