



 
Hi, I'll start the problem for you. The region to be rotated about the yaxis is shown in the diagram along with a dark strip at approximately $x$ on the xaxis, of thickness $\Delta x,$ and height $e^x  x^{1/2}.$ When this strip is rotated about the yaxis it produced a cylindrical shell of height $e^x  x^{1/2},$ thickness $\Delta x$ and radius $x.$ The volume of this shell is approximately its circumference times its height times its thickness, that is \[2 \pi x \times (e^x  x^{1/2}) \times \Delta x.\] I hope this helps,  


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