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Integration for volume of elliptical parabaloid?
solve for volume under the elliptical paraboloid z=2x^2+6y^2 and over the rectangle R=[−5,5]x[−2,2].
(its the rectangle part thats really confusing me, I get 296/3)
so shouldnt it just be:
int (2x^2 + 6y^2) dxdy, x=-5 to -2, y=2 to 5? Because that answer is wrong...
2 Answers
- 1 decade agoFavorite Answer
Regarding your additional details:
No. By definition of the cartesian product of two sets,
[-5,5]x[-2,2] = { (x,y) | -5<=x<=5 and -2<=y<=2 },
(where "<=" means "less than or equal to".)
What do you think that the answer is?
There are sometimes errors in the back of the book...
I came up with 2960/3, so either you have a typo above, or one of us made an arithmetic error... For the rectangle part: The [-5,5] says that x ranges from -5 to 5, and the [-2,2] says that y ranges from -2 to 2. You can set your iterated integral as a dydx integral or as a dxdy integral. If you use dydx, then the "inner" integral sign is the one with -2 and 2 (the y's), and the "outer" integral sign is the one with the -5 and 5 (the x's). For the dydx integral, they switch. Either order should give the same result by Fubini's theorem.
- 1 decade ago
The rectangle defines the limits of integration for the triple integral.
