MATH 241 Lecture Notes - Lecture 8: Multiple Integral

• School: University of Maryland
• Department: Mathematics
• Course: MATH 241
• Professor: Roohollah Ebrahimian

Document Summary

Now we will be looking at double integrals over general regions without the limits being given to us. Recall that the double integral gives the volume under a particular surface and is equal to. This is already set up to be solved so let"s begin: (cid:1516) (cid:1516) (cid:885)(cid:1876)+(cid:888)(cid:1876)(cid:1877) (cid:1876)(cid:1877) (cid:2873)(cid:2870) (cid:3052)4(cid:3052)2. And now we"ll integrate with respect to (cid:1877): find (cid:1517) (cid:887)(cid:1876)(cid:1877) (cid:1876)(cid:2870) where d is the region bounded by (cid:1877)=(cid:1876)(cid:2870) and (cid:1877)= (cid:1876) Since we don"t have bounds for the limits of integration, we will need to sketch the region. And the area of the intersection is the region we are working with. To see that, we will have to set the two equations equal to each other. (cid:1876)(cid:2870)= (cid:1876) (cid:1876)(cid:2872) (cid:1876)=(cid:882) (cid:1876)(cid:4666)(cid:1876)(cid:2871) (cid:883)(cid:4667)=(cid:882) (cid:1876)=(cid:882),(cid:883) (cid:883)(cid:883)(cid:888)(cid:889) way. So the limits of (cid:1876) are from 0 to 1. Those are the limits of (cid:1877) by the. Obviously (cid:1876) grows faster so the limits of (cid:1877) are (cid:1876)(cid:2870) (cid:1877) (cid:1876).