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Well actually I hope the error can be found, so the correct answer can be found

revolving to x-axis yes.

^_^ ty all in advance

"...the surface obtained by rotating the given curves about the x-axis."

your dx/dt is wrong. Look at it again.

The 4 goes to 0, yes?

Yes and in total,
dx/dt = (t+1)e^t

|dw:1341528724406:dw|

|dw:1341528864848:dw|

there is no close solution for integral.

So your answer isn't correct.