This example is from MIT OCW, Multivariable Calculus (http://ocw.mit.edu/courses/mathemati...C_notes_25.pdf)
Shouldn't the highlighted integral start from a, and not zero?
This example is from MIT OCW, Multivariable Calculus (http://ocw.mit.edu/courses/mathemati...C_notes_25.pdf)
Shouldn't the highlighted integral start from a, and not zero?
In polar coordinates, when a point is represented as , r is always the distance from the origin to that point. Here, integrating in polar coordinates, as goes from 0 to , the corresponding r integral is from the origin to the point where the ray at angle crosses the boundary.
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 from the origin. So, the integral should start from zero and go to . As 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.