)yopy said:
I don't think so. The limits of integration in the original problem indicate that the region is to the right of the parabola, not above it. The region is bounded by the parabola, the x-axis, and the line x = 1. Yopy's revised limits of integration describe this region.tiny-tim said:Hi yopy!
(try using the X2 tag just above the Reply box
Well, that looks right to me …
it's just the area above the parabola y = x2 …
Mark44 said:
i should have drawn it! Changing the bounds of integration in a double integral allows you to integrate over non-rectangular regions, making it easier to solve complex problems. It also allows you to switch between Cartesian and polar coordinate systems.
The new bounds of integration can be determined by setting up the appropriate equations for the new coordinate system and solving for the corresponding limits. For example, in polar coordinates, the bounds for r would be determined by the region of integration and the bounds for θ would depend on the geometry of the region.
Yes, the bounds of integration can be changed for any type of double integral, including definite and indefinite integrals. However, it is important to ensure that the new bounds correspond to the same region of integration in both coordinate systems.
Using a change of coordinates can simplify the integration process, especially for complex regions of integration. It can also lead to more elegant solutions and provide insight into the geometry of the problem.
One limitation is that the new coordinate system must be appropriate for the region of integration. For example, using polar coordinates for a rectangular region may not yield the most efficient solution. Additionally, some regions may require multiple changes of coordinates to fully simplify the problem.