A bounded region around Y axis

[Brain0991]

A region is bounded by y=4-e^x, x=0 and y=0 and revolved around the y-axis

I am using the shell method:
2? ?p(x)h(x)dx from a to b

2? ?x(4-e^x)dx
2? ?4x dx - ?xe^x dx
2?[2x^2 - e^x(x-1)]

where I am having the problem is finding the the upper limit.

I know the lower limit is 0 but I get the upper limit to be around 1.4 for some reason this does not seem right to me.

 

[galactus]

The upper limit is 2ln(2)= about 1.4

That is correct. You're OK.

 

[BigGlenntheHeavy]

\(\displaystyle Disc: \ \pi\int_{0}^{3}(ln|4-y|)^2dy \ \dot= \ 8.15838 \ cu. \ units, \ revolved \ around \ y-axis\)

\(\displaystyle Shell: \ 2\pi\int_{0}^{ln(4)}x(4-e^x)dx \ \dot= \ 8.15838 \ cu. \ units, \ revolved \ around \ y-axis\)

\(\displaystyle See \ graph \ below\)

[attachment=0:39trub46]ddd.jpg[/attachment:39trub46]