Find the volume of the solid obtained by rotating the area enclosed by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=5-x }
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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| 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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| then by moving all terms to the left hand side and factoring,
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\pi \int_{x_1}^{\, x_2} R^2-r^2 \,dx}
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\pi \int_{-4}^{\, 5} (25-x^2)^2-(5-x)^2 \,dx}
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\pi \int_{-4}^{\, 5} 625-50x^2+x^4-(25-10x+x^2) \,dx}
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\pi \int_{-4}^{\, 5} 600-51x^2+x^4+10x \,dx}
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\pi \int_{-4}^{\, 5} 600-51x^2+x^4+10x \,dx}
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{15,309}{5} \,\pi.}
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