Saturday, December 21, 2013

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 60

Find the volume generated by rotating the region bounded by y=ex, x=0 and y=π about y-axis. Use cylindrical shells method.
By using horizontal strips, notice that the strips have distance from the x-axis as y such that if you rotate these lengths about the x-axis, you'll have a circumference of c=2πy. Also the height of the strips resembles the height of the cylinder as... H=xrightxleft=lny0=lny. Thus, the volume is...


V=π1c(y)H(y)dyV=π12πy(lny)dyV=2ππ1ylnydy


By using integration by parts, if we let u=lny and dv=ydy, then,
du=1ydy and v=y22

V=2ππ1ylnydy=uvvdu=2π[lny(y22)y22(dyy)]=2π[y22lny12y22]=2πy22[lny12]=πy2[lny12]


Evaluating for y=1 to y=π

=π(π)2[lnπ12]π(1)2[ln(1)12]=π3[lnπ12]+π2 cubic units

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