VOL CROSS-SECTION DISK WASHER

Question

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=e^x, y=0,x= −1, and x=2 about the x-axis.

Answer 1

USING THE WASHER METHOD

V = π∫_-1² [e^x]² dx =(π/2) [ e^2x] |_-1²= (π/2) ( e⁴-e^-2)

USING THE SHELL METHOD

V = ∫_e^-1^{e^2}[ 2πy( 2-lny)] dy + ∫^{e^-1}₀ 2πy( 2-(-1)) dy =

∫_e^-1^{e^2}[ 4π] ydy - ∫_e^-1^{e^2}[ 2πylny] dy + ∫^{e^-1}₀ 6πy dy =

2πe⁴ -2πe^-2 - π ∫_e^-1^{e^2}lnyd y² + 3πe^-2 =

2πe⁴ -2πe^-2 - [πy²lny] |_e^-1^{e^2} + π ∫_e^-1^{e^2} y² (1/y) d y +3πe^-2 =

2πe⁴ -2πe^-2 -πe⁴lne² + πe^-2lne^-1+ π ∫_e^-1^{e^2} y d y +3πe^-2

π/2 [ e⁴-e^-2]