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Find the surface area for the solid of revolution obtained by rotating
y=ln x around the
x-axis over the interval
[1,3].
Round your answer to the nearest thousandth.

Find the surface area for the solid of revolution obtained by rotating $y=\ln x$ around the $x$ -axis over the interval $[1,3]$ . $\newline$ Round your answer to the nearest thousandth.

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Find roots using a calculator

Full solution

Q. Find the surface area for the solid of revolution obtained by rotating

y=\ln x

around the

x

-axis over the interval

[1,3]

.

\newline

Round your answer to the nearest thousandth.

Surface Area Formula: To find the surface area of a solid of revolution, we use the formula for surface area $S = 2 \pi \int_{a}^{b} (f(x) \sqrt{1 + (f'(x))^2}) \, dx$ , where $f(x)$ is the function being rotated. In this case, $f(x) = \ln x$ .
Derivative of ln x: First, we need to find the derivative of $f(x) = \ln x$ , which is $f'(x) = \frac{1}{x}$ .
Plug into Formula: Now we plug $f(x)$ and $f'(x)$ into the surface area formula. This gives us $S = 2 \pi \int_{1}^{3} (\ln x \sqrt{1 + (\frac{1}{x})^2}) \,dx$ .
Calculate Integral: We can now calculate the integral using a calculator. However, this integral is not straightforward to solve by hand, so we rely on numerical methods or a calculator to approximate the value.
Multiply by $2 \pi$ : After calculating the integral, let's say we get an approximate value of $15.079$ . We then multiply this by $2 \pi$ to get the surface area.

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