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=xright−xleft=lny−0=lny. Thus, the volume is...
V=∫π1c(y)H(y)dyV=∫π12πy(lny)dyV=2π∫π1ylnydy
By using integration by parts, if we let u=lny and dv=ydy, then,
du=1ydy and v=y22
V=2π∫π1ylnydy=uv−∫vdu=2π[lny(y22)−∫y22(dyy)]=2π[y22lny−12y22]=2πy22[lny−12]=πy2[lny−12]
Evaluating for y=1 to y=π
=π(π)2[lnπ−12]−π(1)2[ln(1)−12]=π3[lnπ−12]+π2 cubic units
Saturday, December 21, 2013
Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 60
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