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$Find the volume of the solid generated by revolving the regions bounded by the lines and curves y=e^((-1//8)x),y=0, x=0 and x=8 about the x-axis. The volume of the resulting solid is ◻ units cubed. (Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.)$

Find the volume of the solid generated by revolving the regions bounded by the lines and curves $\mathrm{y}=e^{(-1 / 8) \mathrm{x}}, \mathrm{y}=0$ , $x=0$ and $x=8$ about the $x$ -axis. $\newline$ The volume of the resulting solid is $\square$ units cubed. $\newline$ (Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

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Algebra 2

Conjugate root theorems

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Q. Find the volume of the solid generated by revolving the regions bounded by the lines and curves

\mathrm{y}=e^{(-1 / 8) \mathrm{x}}, \mathrm{y}=0

x=0

and

x=8

about the

x

-axis.

\newline

The volume of the resulting solid is

\square

units cubed.

\newline

(Type an exact answer, using

\pi

as needed. Use integers or fractions for any numbers in the expression.)

Identify Function and Limits: Identify the function and limits for the volume integral. We are revolving around the x-axis, so we use the formula for the volume $V$ of a solid of revolution: $V = \pi \int_{a}^{b} (f(x))^2 \, dx$ . Here, $f(x) = e^{(-\frac{1}{8})x}$ , and the limits are from $x = 0$ to $x = 8$ .
Set Up Integral: Set up the integral for the volume. The integral becomes $V = \pi \int_{0}^{8} (e^{(-\frac{1}{8}x)})^2 dx$ . Simplifying the function inside the integral, we get $(e^{(-\frac{1}{8}x)})^2 = e^{(-\frac{1}{4}x)}$ . So, the integral is $V = \pi \int_{0}^{8} e^{(-\frac{1}{4}x)} dx$ .
Calculate Integral: Calculate the integral using the antiderivative. The antiderivative of $e^{(-\frac{1}{4})x}$ is $-4e^{(-\frac{1}{4})x}$ . Evaluating from $0$ to $8$ , we get: $\newline$ $[-4e^{(-\frac{1}{4})x}]$ from $0$ to $8$ = $[-4e^{(-\frac{1}{4})\cdot 8}] - [-4e^0]$ = $-4e^{-2} - (-4)$ = $4 - 4e^{-2}$ .
Multiply by $\pi$ : Multiply by $\pi$ to find the volume. The volume $V = \pi(4 - 4e^{-2})$ .