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I'm trying to evaluate: $\int\int xydA$

Where D is the region bounded by the line $y=x-2$ and $x=y^2$

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Does the integral need to be set up as:

$$\int_{-1}^{2} \int_{y+2}^{y^2} xydxdy$$

or do I need to evaluate the double integral of the area above the y-axis and below separately and add them together like this?

$$\int_{-1}^{0} \int_{x-2}^{\sqrt x} xydydx + \int_{0}^{2} \int_{x-2}^{\sqrt x} xydydx$$

Or am I completely wrong and it's something totally different?

z400jt
  • 127

1 Answers1

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The two integrals are the same. In this case you do not need to break it up (though you're welcome to if you want to). In some integrals you have to break it up because the bounds change. If your bounding functions cross, or if you have several functions bounding a shape and which one is bounding the slice you are looking at changes, then you have to break it up because the integrals can't combine the way these two can.

Stella Biderman
  • 31,155