Find the volume of the solid obtained by rotating the area enclosed by 
and 
around the x-axis.
| Foundations
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| • Choose either shell or washer method.
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| • Find the appropriate radii.
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| • Determine the bounds of integration by finding when both functions have the same y value.
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| • Using the determined values, set up and solve the integral.
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Solution:
| Step 1:
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| Choosing the Approach: Since we are rotating around the x-axis, the washer method would utilize tall rectangles with dx as their width. This seems like a reasonable choice, as these rectangles would be trapped between our two functions as x varies over the enclosed region, allowing us to solve a single integral. Note that the washer method will require an inner and outer radius, as well as bounds of integration, in order to evaluate the integral
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V=\pi \int _{x_{1}}^{\,x_{2}}R^{2}-r^{2}\,dx}
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| Step 2:
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| Finding the Radii: Since our rectangles will be trapped between the two functions, and will be rotated around the x-axis (where y = 0), we find
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the inner radius is  , represented by the blue line, while
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the outer radius is  , represented by the red line.
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| Step 3:
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| Finding the Bounds of Integration: We must set the two functions equal, and solve. If
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 5-x=25-x^{2},}
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| then by moving all terms to the left hand side and factoring,
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{2}-x-20=(x+4)(x-5)=0,}
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| so we have -4 and 5 as solutions. These are our bounds of integration.
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| Step 4:
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| Evaluating the Integral: Using the earlier steps, we have
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V=\pi \int _{x_{1}}^{\,x_{2}}R^{2}-r^{2}\,dx}
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\pi \int _{-4}^{\,5}(25-x^{2})^{2}-(5-x)^{2}\,dx}
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\pi \int _{-4}^{\,5}625-50x^{2}+x^{4}-(25-10x+x^{2})\,dx}
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\pi \int _{-4}^{\,5}600-51x^{2}+x^{4}+10x\,dx}
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\pi {\biggr (}600x-51\cdot {\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+10\cdot {\frac {x^{2}}{2}}{\biggr )}{\biggr |}_{x=-4}^{5}}
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ={\frac {15,309}{5}}\,\pi .}
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