Volume of Revolution & Kinematics (HL)
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
Find the volume when \(y=\dfrac{1}{x}\) for \(1\leq x\leq4\) is rotated \(2\pi\) about the \(x\)-axis.
Formula
\(V=\pi\displaystyle\int_1^4\dfrac{1}{x^2}\,dx=\pi[-x^{-1}]_1^4=\pi(- frac{1}{4}+1)\)
Answer
\(\dfrac{3\pi}{4}\) cubic units
Question 2
The region bounded by \(y=\ln x\), \(x=1\), \(x=e\), and the \(x\)-axis is rotated \(2\pi\) about the \(x\)-axis. Find the volume.
Formula
\(V=\pi\displaystyle\int_1^e(\ln x)^2\,dx\); use IBP with \(u=(\ln x)^2,\;dv=dx\)
After IBP
\([x(\ln x)^2]_1^e-2\displaystyle\int_1^e\ln x\,dx=e-2[x\ln x-x]_1^e=e-2(e-e+1)=e-2\)
Answer
\(\pi(e-2)\) cubic units
Question 3
A particle moves along a straight line with acceleration \(a(t)=6t-4\) m/s². At \(t=0\): \(v=2\) m/s and \(s=0\) m.
(a)Find \(v(t)\) and \(s(t)\).
(b)Find the total distance travelled for \(0\leq t\leq3\).
(a)
Integrate \(a\)
\(v=3t^2-4t+C_1\); \(v(0)=2\Rightarrow C_1=2\). \(s=t^3-2t^2+2t+C_2\); \(s(0)=0\Rightarrow C_2=0\)
Answer
\(v(t)=3t^2-4t+2\); \(s(t)=t^3-2t^2+2t\)
(b)
Check if \(v=0\)
Discriminant of \(3t^2-4t+2\): \(16-24<0\) — no real roots, so \(v>0\) always
Distance equals displacement
\(s(3)=27-18+6=15\)
Question 4
Find the volume generated when \(y= an x\) for \(0\leq x\leq\dfrac{\pi}{4}\) is rotated \(2\pi\) about the \(x\)-axis.
Formula
\(V=\pi\displaystyle\int_0^{\pi/4} an^2x\,dx=\pi\displaystyle\int_0^{\pi/4}(\sec^2x-1)\,dx\)
Integrate
\(\pi[ an x-x]_0^{\pi/4}=\pi(1- frac{\pi}{4})\)
Answer
\(\pi\!\left(1-\dfrac{\pi}{4}
ight)\) cubic units
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