Double Integral Question

• Jul 16th 2014, 10:07 AM
cosmonavt
Double Integral Question
This example is from MIT OCW, Multivariable Calculus (http://ocw.mit.edu/courses/mathemati...C_notes_25.pdf)

Attachment 31300

Shouldn't the highlighted integral start from a, and not zero?
• Jul 16th 2014, 10:43 AM
HallsofIvy
Re: Double Integral Question
In polar coordinates, when a point is represented as $(r, \theta)$, r is always the distance from the origin to that point. Here, integrating in polar coordinates, as $\theta$ goes from 0 to $\pi/4$, the corresponding r integral is from the origin to the point where the ray at angle $\theta$ crosses the boundary.
• Jul 16th 2014, 10:49 AM
SlipEternal
Re: Double Integral Question
At the origin (which is part of R), the distance to the origin is zero. The point on the edge of the square is a distance of $a\sec \theta$ from the origin. So, the integral should start from zero and go to $a\sec \theta$. As $\theta$ changes, it represents a different line segment of points in that triangle. The first line segment is from the origin along the x-axis to the edge of the square. The last one is the hypotenuse of the triangle.
• Jul 17th 2014, 12:57 AM
cosmonavt
Re: Double Integral Question
I got that. Thank you very much both of you.