Section 8.2 : Surface Area
5. Find the surface area of the object obtained by rotating y=sin(2x) , 0≤x≤π8 about the x-axis.
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Start SolutionNote that we actually set this problem up in Part (a) of Problem 2. So, we’ll just summarize the steps of the set up part of the problem here. If you need to see all the details please check out the work in Problem 2.
Here is ds for this problem.
dydx=2cos(2x)⇒ds=√1+4cos2(2x)dxThe integral for the surface area is,
SA=π8∫02πsin(2x)√1+4cos2(2x)dx Show Step 2In order to evaluate this integral we’ll need the following trig substitution.
cos(2x)=12tan(θ)→−2sin(2x)dx=12sec2(θ)dθ√1+4cos2(2x)=√1+tan2(θ)=√sec2(θ)=|sec(θ)|In order to deal with the absolute value bars we’ll need to convert the x limits to θ limits. Here’s that work.
x=0:cos(0)=1=12tan(θ)→θ=tan−1(2)=1.1071x=π8:cos(π4)=√22=12tan(θ)→θ=tan−1(√2)=0.9553The corresponding range of θ is 0.9553≤θ≤1.1071 . This is in the first quadrant and secant is positive there. Therefore, we can drop the absolute value bars on the secant.
Show Step 3Putting all the work from the previous step together gives,
SA=π8∫02πsin(2x)√1+4cos2(2x)dx=−π2∫0.95531.1071sec3(θ)dθ Show Step 4Using the formula for the integral of sec3(θ) we derived in the Integrals Involving Trig Functions we get,
SA = - \frac{\pi }{2}\int_{{1.1071}}^{{0.9553}}{{{{\sec }^3}\left( \theta \right)\,d\theta }} = \left. { - \frac{\pi }{4}\left[ {\sec \left( \theta \right)\tan \left( \theta \right) + \ln \left| {\sec \left( \theta \right) + \tan \left( \theta \right)} \right|} \right]} \right|_{1.1071}^{0.9553} = \require{bbox} \bbox[2pt,border:1px solid black]{{1.8215}}Note that depending upon the number of decimal places you used your answer may be slightly different from that give here. The “exact” answer, obtained by computer, is 1.8222.