Question

The region between the curve y=1/(2x)y=1 /(2 \sqrt{x}) and the x-axis from x = 1/4 to x = 4 is revolved about the x-axis to generate a solid.

Find the volume of the solid.

Solution

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Answered 4 months ago
Answered 4 months ago
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'slader'

Using disk method and vertical strip, the volume is:

V=abπ[R(x)]2dx=π1/4414xdx=π41/441xdx=π4lnx1/44=π4(ln4ln14)=π4(ln4+ln4)=π2ln4=πln41/2=πln2\begin{aligned} V&=\int_a^b \pi [R(x)]^2\: dx \\ &=\pi \int_{1/4}^4 \frac{1}{4x}\: dx \\ &=\frac{\pi}{4}\int_{1/4}^4 \frac{1}{x}\: dx \\ &=\frac{\pi}{4}\left.\ln x\right|_{1/4}^4 \\ &=\frac{\pi}{4}\left(\ln 4-\ln \frac{1}{4}\right) \\ &=\frac{\pi}{4}(\ln 4+\ln 4) \\ &=\frac{\pi}{2}\ln 4 \\ &=\pi \ln 4^{1/2} \\ &=\pi \ln 2 \end{aligned}

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