# Parametric equation and limits

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.

• It depends on what region are you integrating. The limits can't be decided just by the function. We need some more information. – HS Singh Aug 12 at 12:25
• We are integrating in entire region of circle . – shubham jain Aug 12 at 12:26
• Can t be ever equal to π ?.At π circle won't be there .Circle is there in 1st and 4th quadrant. – shubham jain Aug 12 at 12:28
• Can you upload the exact question? – HS Singh Aug 12 at 12:31
• Ok so question is calculate the volume bounded by Sphere x^2 +y^2 + z^2 =a^2 and Cylinder x^2 +y ^2 = 2ax .Please try to solve it in parametric form . – shubham jain Aug 12 at 12:40

## 1 Answer

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 ,please if you evaluate it completely .I am still having some doubts . – shubham jain Aug 12 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 at 15:44
• math.stackexchange.com/questions/3321156/… 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 at 15:48
• Result in the link does not seem right, which ought to have $\pi$. – Quanto Aug 12 at 17:55
• Its a printing error in last step .Anyways , satisfactory answer has been given by Hans.Thanks – shubham jain Aug 12 at 18:03