Find the volume of the solid

Question

Find the volume of the solid of revolution obtained by rotating the region bounded by the curve y=4−e^x and the line x=0 and y=0 about the line x = −3. Sketch the region and the axis of revolution for full credit.

Answer 1

Please see the video.

Answer 2

As we are rotating about x and the functions are in terms of x, we will use the shell method. As we are rotating about x = -3, the radius is x+ 3; the height is 4 - e^x; 4 - e^x = 0 when e^x = 4, or x = ln(4)

Thus, we integrate 2π (x+3)(4-e^x) from 0 to ln 4

I[0, ln 4] 2π (x+3)(4-e^x) dx = 2π I[0, ln 4] 4x+12 - xe^x - 3e^x) dx = 2π (2x^2 + 12x - xe^x - 2 e^x)E[0, ln 4] =

2π (2 (ln 4)^2 + 12 ln(4) - 4 ln 4 - 2 * 4 + 2) =

4π (ln 4)^2 + 16π ln 4 - 12π =

56.1338

Answer 3

desmos.com/calculator/aufo1dux4r