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I have a fundamental doubt .Suppose , we are integrating a double integral in X-Y plane , with parametric equation reducing to $$ r= a \cos{t} $$.Then , I am of the view that limits of t should ideally vary from -π/2 to +π /2 .But , there are many text books ,taking the limits for t ( while doing the double integral ) as 0 to π .Is it justified .At π ,radius vector is negative ,so how can limit vary till π.Please ,if anyone can justify.

Matti P.
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shubham jain
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1 Answers1

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For your particular example, i.e. the volume enclosed by a sphere and a cylinder, the parametric integral should take the form

$$\int_{0}^{\pi}\sin\theta d\theta\int_{-\pi/2}^{\pi/2} d\phi\int_0^{f(\theta,\phi)}r^2dr$$

The range for either angle integration is $\pi$.

Quanto
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  • Quanto ,please if you evaluate it completely .I am still having some doubts . – shubham jain Aug 12 '19 at 13:20
  • The actual integration is quite involved due to overlapping between the two shapes. I'll work it out when I get a chance. – Quanto Aug 12 '19 at 15:44
  • https://math.stackexchange.com/questions/3321156/limits-of-integration-for-parametric-equation.Just now i have added solution to that question .Please visit the link .I have explicitly mentioned my exact doubt there .Please see if you can help me out . – shubham jain Aug 12 '19 at 15:48
  • Result in the link does not seem right, which ought to have $\pi$. – Quanto Aug 12 '19 at 17:55
  • Its a printing error in last step .Anyways , satisfactory answer has been given by Hans.Thanks – shubham jain Aug 12 '19 at 18:03