Question

Asked May 14, 2019

Define R as the region bounded by the graphs of f(x)=x^{2} +4x+5, x=−3, x=0, and the x-axis. Using the disk method, what is the volume of the solid of revolution generated by rotating R about the x-axis? Enter your answer in terms of π.

Step 1

Please see the diagram on the white board. The line in blue is the plot of the given curve f(x)=x^{2} +4x+5 from x=−3 to x=0.

A disc is shown at distance x with radius R = y and thickness dx.

The volume of this disc = dV = πR^{2}dx = πy^{2}dx = π(x^{2} +4x+5)^{2}dx = π(x^{4} + 8x^{3} + 26x^{2 }+ 40x + 25)dx

Step 2

So, the desired volume is the integral of dV over x = - 3 to x = 0.

Recall the famous...

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