# Prototype Calculus Question

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Find the volume of the solid obtained by rotating the area enclosed by ${\displaystyle y=5-x}$ and ${\displaystyle y=25-x^{2}}$
around the x-axis.

Foundations
• Choose either shell or washer method.
• Find the appropriate radii.
• Determine the bounds of integration by finding when both functions have the same y value.
• Using the determined values, set up and solve the integral.

Solution:

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