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Find the volume of the solid obtained when the given region is rotated about the x-axis.
Under [itex]y = (\sin{x})^{\frac{3}{2}}[/itex] between 0 and pi.
The radius is... [itex]r = (\sin{x})^{\frac{3}{2}}[/itex]
Then the area for any sample is... [itex]A (x) = \pi((\sin{x})^{\frac{3}{2}})^{2}[/itex]
Simplifying to... [itex]A (x) = \pi(\sin{x})^{3}[/itex]
Integrate between 0 and pi to get the volume...
[tex]V = \pi \int_{0}^{\pi} (\sin{x})^{3}[/tex]
[tex]V = \pi [ \frac{(\sin{x})^{4}}{4\cos{x}} ]_{0}^{\pi}[/tex]
But... sin(pi) and sin(0) both equal 0, making the volume 0. But it's actually (4/3)(pi). What am I missing?
Under [itex]y = (\sin{x})^{\frac{3}{2}}[/itex] between 0 and pi.
The radius is... [itex]r = (\sin{x})^{\frac{3}{2}}[/itex]
Then the area for any sample is... [itex]A (x) = \pi((\sin{x})^{\frac{3}{2}})^{2}[/itex]
Simplifying to... [itex]A (x) = \pi(\sin{x})^{3}[/itex]
Integrate between 0 and pi to get the volume...
[tex]V = \pi \int_{0}^{\pi} (\sin{x})^{3}[/tex]
[tex]V = \pi [ \frac{(\sin{x})^{4}}{4\cos{x}} ]_{0}^{\pi}[/tex]
But... sin(pi) and sin(0) both equal 0, making the volume 0. But it's actually (4/3)(pi). What am I missing?