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roisin's question at Yahoo! Answers regarding the volume of a torus
- Thread starter MarkFL
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Hello roisin,
The equation describing the circle to be rotated is:
\(\displaystyle (x-b)^2+y^2=a^2\)
Now, so that we know what our goal is, using the formula for the volume of a torus, we should expect to find the volume of the solid of revolution to be:
\(\displaystyle V=2\pi^2a^2b\)
Let's use both the washer and shell methods. We'll begin with the washer method.
Washer Method
The volume of an arbitrary washer is:
\(\displaystyle dV=\pi\left(R^2-r^2 \right)\,dy\)
where:
\(\displaystyle R=b+\sqrt{a^2-y^2}\)
\(\displaystyle r=b-\sqrt{a^2-y^2}\)
Hence:
\(\displaystyle R^2-r^2=(R+r)(R-r)=(2b)\left(2\sqrt{a^2-y^2} \right)=4b\sqrt{a^2-y^2}\)
And so the volume of the arbitrary washer can be written as:
\(\displaystyle dV=4\pi b\sqrt{a^2-y^2}\,dy\)
Now, summing the washers, we get the volume of the solid:
\(\displaystyle V=4\pi b\int_{-a}^a \sqrt{a^2-y^2}\,dy\)
Using the even-function rule, this becomes:
\(\displaystyle V=8\pi b\int_0^a \sqrt{a^2-y^2}\,dy\)
Using the substitution:
\(\displaystyle y=a\sin(\theta)\,\therefore\,dy=a\cos(\theta)\,d \theta\)
We have:
\(\displaystyle V=8\pi b\int_{\theta(0)}^{\theta(a)}\sqrt{a^2-a^2\sin^2(\theta)}\,a\cos(\theta)\,d\theta\)
Now, to change the limits of integration, observe we have:
\(\displaystyle \theta(x)=\sin^{-1}\left(\frac{y}{a} \right)\) and so:
\(\displaystyle \theta(0)=\sin^{-1}\left(\frac{0}{a} \right)=0\)
\(\displaystyle \theta(a)=\sin^{-1}\left(\frac{a}{a} \right)=\frac{\pi}{2}\)
Now, on the interval \(\displaystyle \left(0,\frac{\pi}{2} \right)\), we have the sine and cosine functions being non-negative, hence we may write the integral as:
\(\displaystyle V=8\pi b\int_{0}^{\frac{\pi}{2}}a\sqrt{1-\sin^2(\theta)}\,a\cos(\theta)\,d\theta\)
\(\displaystyle V=8\pi a^2b\int_{0}^{\frac{\pi}{2}}cos^2(\theta)\,d\theta\)
Now, using the identity \(\displaystyle \cos^2(x)=\frac{1+\cos(2x)}{2}\) we have:
\(\displaystyle V=4\pi a^2b\int_{0}^{\frac{\pi}{2}}1+\cos(2\theta)\,d \theta\)
Hence:
\(\displaystyle V=4\pi a^2b\left[\theta+\frac{1}{2}\sin(2\theta) \right]_{0}^{\frac{\pi}{2}}=4\pi a^2b\left(\frac{\pi}{2} \right)=2\pi^2a^2b\)
Okay, this is the result we expected. Now let's look at the shell method.
Shell Method
The volume of an arbitrary shell is:
\(\displaystyle dV=2\pi rh\,dx\)
where:
\(\displaystyle r=x\)
\(\displaystyle h=2\sqrt{a^2-(x-b)^2}\)
And thus the volume of the arbitrary shell is:
\(\displaystyle dV=4\pi x\sqrt{a^2-(x-b)^2}\,dx\)
Summing up the shells, we get the volume of the solid:
\(\displaystyle V=4\pi\int_{b-a}^{b+a} x\sqrt{a^2-(x-b)^2}\,dx\)
Using the substitution:
\(\displaystyle x-b=a\sin(\theta)\,\therefore\,dx=a\cos(\theta)\, d\theta\)
we obtain:
\(\displaystyle V=4\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(a\sin(\theta)+b \right)\sqrt{a^2-a^2\sin^2(\theta)}\,a\cos(\theta)\,d\theta\)
\(\displaystyle V=4\pi a^2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(a\sin(\theta)+b \right)\sqrt{1-\sin^2(\theta)}\,\cos(\theta)\,d\theta\)
\(\displaystyle V=4\pi a^2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(a\sin(\theta)+b \right)\cos^2(\theta)\,d\theta\)
\(\displaystyle V=4\pi a^3\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin(\theta)\cos^2(\theta)\,dx+4\pi a^2b\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2(\theta)\, d\theta\)
Using the odd and even function rules, we obtain:
\(\displaystyle V=8\pi a^2b\int_{0}^{\frac{\pi}{2}}\cos^2(\theta)\, d\theta\)
Now, using the identity \(\displaystyle \cos^2(x)=\frac{1+\cos(2x)}{2}\) we have:
\(\displaystyle V=4\pi a^2b\int_{0}^{\frac{\pi}{2}}1+\cos(2\theta)\,d \theta\)
Hence:
\(\displaystyle V=4\pi a^2b\left[\theta+\frac{1}{2}\sin(2\theta) \right]_{0}^{\frac{\pi}{2}}=4\pi a^2b\left(\frac{\pi}{2} \right)=2\pi^2a^2b\)
Here we have also obtained the desired result.
The equation describing the circle to be rotated is:
\(\displaystyle (x-b)^2+y^2=a^2\)
Now, so that we know what our goal is, using the formula for the volume of a torus, we should expect to find the volume of the solid of revolution to be:
\(\displaystyle V=2\pi^2a^2b\)
Let's use both the washer and shell methods. We'll begin with the washer method.
Washer Method
The volume of an arbitrary washer is:
\(\displaystyle dV=\pi\left(R^2-r^2 \right)\,dy\)
where:
\(\displaystyle R=b+\sqrt{a^2-y^2}\)
\(\displaystyle r=b-\sqrt{a^2-y^2}\)
Hence:
\(\displaystyle R^2-r^2=(R+r)(R-r)=(2b)\left(2\sqrt{a^2-y^2} \right)=4b\sqrt{a^2-y^2}\)
And so the volume of the arbitrary washer can be written as:
\(\displaystyle dV=4\pi b\sqrt{a^2-y^2}\,dy\)
Now, summing the washers, we get the volume of the solid:
\(\displaystyle V=4\pi b\int_{-a}^a \sqrt{a^2-y^2}\,dy\)
Using the even-function rule, this becomes:
\(\displaystyle V=8\pi b\int_0^a \sqrt{a^2-y^2}\,dy\)
Using the substitution:
\(\displaystyle y=a\sin(\theta)\,\therefore\,dy=a\cos(\theta)\,d \theta\)
We have:
\(\displaystyle V=8\pi b\int_{\theta(0)}^{\theta(a)}\sqrt{a^2-a^2\sin^2(\theta)}\,a\cos(\theta)\,d\theta\)
Now, to change the limits of integration, observe we have:
\(\displaystyle \theta(x)=\sin^{-1}\left(\frac{y}{a} \right)\) and so:
\(\displaystyle \theta(0)=\sin^{-1}\left(\frac{0}{a} \right)=0\)
\(\displaystyle \theta(a)=\sin^{-1}\left(\frac{a}{a} \right)=\frac{\pi}{2}\)
Now, on the interval \(\displaystyle \left(0,\frac{\pi}{2} \right)\), we have the sine and cosine functions being non-negative, hence we may write the integral as:
\(\displaystyle V=8\pi b\int_{0}^{\frac{\pi}{2}}a\sqrt{1-\sin^2(\theta)}\,a\cos(\theta)\,d\theta\)
\(\displaystyle V=8\pi a^2b\int_{0}^{\frac{\pi}{2}}cos^2(\theta)\,d\theta\)
Now, using the identity \(\displaystyle \cos^2(x)=\frac{1+\cos(2x)}{2}\) we have:
\(\displaystyle V=4\pi a^2b\int_{0}^{\frac{\pi}{2}}1+\cos(2\theta)\,d \theta\)
Hence:
\(\displaystyle V=4\pi a^2b\left[\theta+\frac{1}{2}\sin(2\theta) \right]_{0}^{\frac{\pi}{2}}=4\pi a^2b\left(\frac{\pi}{2} \right)=2\pi^2a^2b\)
Okay, this is the result we expected. Now let's look at the shell method.
Shell Method
The volume of an arbitrary shell is:
\(\displaystyle dV=2\pi rh\,dx\)
where:
\(\displaystyle r=x\)
\(\displaystyle h=2\sqrt{a^2-(x-b)^2}\)
And thus the volume of the arbitrary shell is:
\(\displaystyle dV=4\pi x\sqrt{a^2-(x-b)^2}\,dx\)
Summing up the shells, we get the volume of the solid:
\(\displaystyle V=4\pi\int_{b-a}^{b+a} x\sqrt{a^2-(x-b)^2}\,dx\)
Using the substitution:
\(\displaystyle x-b=a\sin(\theta)\,\therefore\,dx=a\cos(\theta)\, d\theta\)
we obtain:
\(\displaystyle V=4\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(a\sin(\theta)+b \right)\sqrt{a^2-a^2\sin^2(\theta)}\,a\cos(\theta)\,d\theta\)
\(\displaystyle V=4\pi a^2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(a\sin(\theta)+b \right)\sqrt{1-\sin^2(\theta)}\,\cos(\theta)\,d\theta\)
\(\displaystyle V=4\pi a^2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(a\sin(\theta)+b \right)\cos^2(\theta)\,d\theta\)
\(\displaystyle V=4\pi a^3\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin(\theta)\cos^2(\theta)\,dx+4\pi a^2b\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2(\theta)\, d\theta\)
Using the odd and even function rules, we obtain:
\(\displaystyle V=8\pi a^2b\int_{0}^{\frac{\pi}{2}}\cos^2(\theta)\, d\theta\)
Now, using the identity \(\displaystyle \cos^2(x)=\frac{1+\cos(2x)}{2}\) we have:
\(\displaystyle V=4\pi a^2b\int_{0}^{\frac{\pi}{2}}1+\cos(2\theta)\,d \theta\)
Hence:
\(\displaystyle V=4\pi a^2b\left[\theta+\frac{1}{2}\sin(2\theta) \right]_{0}^{\frac{\pi}{2}}=4\pi a^2b\left(\frac{\pi}{2} \right)=2\pi^2a^2b\)
Here we have also obtained the desired result.