Help with Integrals

Question

What is the integral that will find the volume of the solid that is formed when the region bounded by the graphs of y = ex, x = 2, and y = 1 is revolved around the line y = -1?

Answer 1

Rotating y=e^x around line y=-1 is the same as rotating y=e^x+1 around x-axis and the volume of the solid of revolution around x-axis is

V=∫_a^bπy²(x)dx.

e^x=1, x=0, So, 0≤x≤2 ad the volume is

V=π∫₀²(e^x+1)²dx=(π/2)(e⁴+4e²-1).