Volume of Revolution
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find the volume of the solid formed when the region bounded by \(y=x^2\), \(x=0\), \(x=2\) and the \(x\)-axis is rotated \(2\pi\) radians about the \(x\)-axis.
Formula
\(V=\pi\displaystyle\int_0^2(x^2)^2\,dx=\pi\displaystyle\int_0^2 x^4\,dx\)
Integrate
\(\pi\left[ frac{x^5}{5}
ight]_0^2=\pi\cdot frac{32}{5}\)
Answer
\(\dfrac{32\pi}{5}\) cubic units
Question 2
Find the volume when \(y=\sqrt{x+1}\) for \(0\leq x\leq3\) is rotated about the \(x\)-axis.
Formula
\(V=\pi\displaystyle\int_0^3(x+1)\,dx=\pi[ frac{x^2}{2}+x]_0^3=\pi( frac{9}{2}+3)\)
Answer
\(\dfrac{15\pi}{2}\) cubic units
Question 3
Find the volume when \(y=e^x\) for \(0\leq x\leq1\) is rotated about the \(x\)-axis.
Formula
\(V=\pi\displaystyle\int_0^1 e^{2x}\,dx=\pi\left[ frac{e^{2x}}{2}
ight]_0^1\)
Answer
\(\dfrac{\pi(e^2-1)}{2}\) cubic units
Question 4
Find the volume when \(y=\sin x\) for \(0\leq x\leq\pi\) is rotated about the \(x\)-axis.
Formula
\(V=\pi\displaystyle\int_0^\pi\sin^2x\,dx=\pi\displaystyle\int_0^\pi frac{1-\cos2x}{2}\,dx\)
Integrate
\(\pi\left[ frac{x}{2}- frac{\sin2x}{4}
ight]_0^\pi=\pi\cdot frac{\pi}{2}\)
Answer
\(\dfrac{\pi^2}{2}\) cubic units
Question 5
The region bounded by \(y=\sqrt{x}\) and \(y=x^2\) (for \(x\geq0\)) is rotated \(2\pi\) about the \(x\)-axis. Find the volume.
Intersections
\(\sqrt{x}=x^2\Rightarrow x=0,1\)
Volume between curves
\(V=\pi\displaystyle\int_0^1[({\sqrt{x}})^2-(x^2)^2]\,dx=\pi\displaystyle\int_0^1[x-x^4]\,dx=\pi[ frac{x^2}{2}- frac{x^5}{5}]_0^1\)
Answer
\(\dfrac{3\pi}{10}\) cubic units
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