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 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=25-x^2 }
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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| Solve the integral.
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Solution:
| Step 1:
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| 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.
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| Step 2:
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| Since our rectangles will be trapped between the two functions, and will be rotated around the x-axis (where 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=0 }
,) we find
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| the inner radius is 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 r = 5-x }
, represented by the blue line, while
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| the outer radius is 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 R = 25-x^2 }
, represented by the red line.
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| Step 3:
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| We must set the two functions equal, and solve. More to follow...
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