Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 2e−x, y = 2, x = 6; about y = 4

Question

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Answer 1

Draw the region.  y = 2e^-x  and y = 4 intersect when 2e^-x = 4.  Solving the equation, we get x = -ln2.

 

Using the disk method, the volume of a typical disk is π(4-2e^-x)²Δx

 

So, the volume of the solid is ∫_{(from -ln2 to 6)}(4-2e^-x)²dx.

 

                                       = ∫_{(from -ln2 to 6)}[16-16e^-x+4e^-2x]dx

 

If you need help in evaluating the integral, let me know.