Where πy² comes from, what changes when you spin about the other axis, and areas measured against the y-axis.
Here is y = x². Switch the axis and watch the solid change completely. The curve has not moved; only what you are spinning it around has.
Each slice is a disc. The only question is what its radius is.
About the x-axis: V = ∫ab πy² dx. The radius is the height y, the slices are vertical, and the limits are x values.
About the y-axis: V = ∫cd πx² dy. The radius is the horizontal distance x, the slices are horizontal, and the limits are y values.
For the y-axis, rearrange first. If the question gives y = x², then x² = y, and the integral of πy dy is straightforward. Trying to integrate x² with respect to y without rearranging is where this goes wrong.
Different numbers, because they are different solids. The first is a bowl shape pointing along the x-axis; the second is the region outside the curve spun into a deeper bowl.
The same swap works for areas. The region between a curve and the y-axis is ∫x dy, with limits in y.
1. Rotate y = x², 0 ≤ x ≤ 2, about the x-axis. Find the volume to 2 decimal places.
2. Rotating about the y-axis, the integrand is:
3. Rotate y = x² for 0 ≤ y ≤ 4 about the y-axis. Find the volume to 2 decimal places.
Writing the correct formula with the correct limits before substituting. Mixing an x limit into a dy integral is the error that this sub-topic exists to catch.
Rearranging the equation into the variable you are integrating with respect to.
Squaring correctly, including any constant term, and keeping the π all the way to the end.
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