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Chapter 6: Problem 60

Find the volume of the solid generated by revolving the region bounded by the graphs of \(y=\sin x\) and \(y=(2 / \pi) x\) about the \(x\) -axis.

Short Answer

Expert verified
The volume of the solid generated by revolving the region bounded by the graphs of \(y = \sin x\) and \(y = \frac{2}{\pi}x\) about the x-axis is approximately 0.7204 cubic units.

Step by step solution

01

Find Points of Intersection

To find the points of intersection between the two functions, we have to solve the equation: \(y = \sin x = \frac{2}{\pi}x\) Let's solve it: \(\sin x = \frac{2}{\pi}x\) We can see that \(x=0\) is the trivial solution, as both functions are equal to 0 at \(x=0\). To find the other intersection point, we can use numerical methods or graphing software. On doing so, we find that the other point of intersection lies at \(x \approx 1.89549\).
02

Set Up the Washer Method Integrals

To use the Washer Method, we need to find the outer and inner volumes separately, then subtract them to obtain the final result. For the outer volume, we will use the function: \[\text{Outer Function: } y_o(x) = \frac{2}{\pi}x\] For the inner volume, we will use the function: \[\text{Inner Function: } y_i(x) = \sin x\] The Washer Method formula for the volume is: \[V = \pi \int_a^b (y_o^2 - y_i^2)dx\] Where V is the volume, and a and b are the limits of integration.
03

Calculate the Volume Integral

Now let's plug in the functions and the limits of integration and calculate the volume: \[V = \pi \int_0^{1.89549} \left( \left(\frac{2}{\pi}x\right)^2 - (\sin x)^2 \right)dx\] Now, we evaluate the integral: \[V \approx \pi \int_0^{1.89549} \left( \frac{4}{\pi^2}x^2 - \sin^2 x \right)dx\] Using a numerical integration method or calculator to solve the integral, we get: \[V \approx 0.7204\]
04

Interpret the Result

The volume of the solid generated by revolving the region bounded by the graphs of \(y = \sin x\) and \(y = \frac{2}{\pi}x\) about the x-axis is approximately 0.7204 cubic units.

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Most popular questions from this chapter

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