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The volume of solid obtained by revolving the area of the ellipse x2a2+y2b2=1 about major and minor axes are in the ratio.

Answer
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Hint: In this question, we are given equation of ellipse as x2a2+y2b2=1. We need to find the ratio of volume of solid obtained by revolving the area of the ellipse about major and minor axes. Center of the ellipse is at origin, so we will find a volume of solid revolving about the x axis and y axis. Volume of any solid revolving an area given by equation g(x,y) about x axis is equal to abπy2dx where y = f(x) and a,b are points on x axis where g(x,y) cuts. Similarly, about y axis volume is equal to abπx2dy where x = f(y) and a,b are points on y axis where g(x,y) cuts.

Complete step by step answer:
Here we are given the equation of the ellipse as x2a2+y2b2=1.
Let us first draw a diagram for this ellipse. We have,
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We need to find the volume of solid obtained by revolving the area of the ellipse about the major axis and minor axis. From the diagram we can see that, the major axis is the x axis and minor axis is the y axis.
We know volume of solid when equation g(x,y) is revolved about x axis is given by v=abπy2dx where y = f(x), a,b are points where g(x,y) cuts x axis.
So let us use this formula, here g(x,y)=x2a2+y2b2=1.
Calculating value of y2 we get,
y2b2=1x2a2y2=b2(1x2a2)y2=b2b2x2a2.
From diagram a,b are -a,a respectively, so we have
v=aaπ(b2b2x2a2)dy.
We know xndx=xn+1n+1 and 1dx=x so we have,
v=[πb2xπb2a2x2+12+1]aav=[πb2xπb2a2x33]aa.
Solving further we get,
v=[πb2aπb2a2a33(πb2(a)πb2a2(a)33)][πb2aπb2a3+πb2aπb2a3]2πb2a2πb2a34πb2a3
Here volume is 4πb2a3 when revolved around the major axis.
When a solid is revolved around y axis we have the formula v=abπx2dy.
Let us find the value of x2.
x2a2=1y2b2x2=a2a2y2b2.
From diagram a,b are -b, b respectively so we have,
v=bbπ(a2a2y2b2)dx.
Solving the integration we have,
v=[πa2ya2y3b23]bbv=[πa2ba2b33b2(πa2(b)a2(b)3b23)][πa2bπa2b3+πa2bπa2b3]2πa2b2πa2b343πa2b.
Hence volume is 43πa2b when revolved around a minor axis.
Now let us take the ratio of the volume about the major axis to the volume about the minor axis. We have
Volume about major axisVolume about minor axis=43πab243πa2b=ba
Hence the required ratio is b:a.

Note: Students should keep in mind the formula of volume of solid when area is revolved about x axis and y axis. Keep in mind the formula of integration. Try to draw a diagram first for finding the required points (a,b). Take care of signs while solving the integration. Do not get confused between the major axis and the minor axis.
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