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Problem 37

Although it is often true that a double integral …

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Problem 37

Although it is often true that a double integral can be evaluated by using either $d x$ or $d y$ first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises 37 and 38 in the easiest way possible.
$$\iint_{R} x e^{x y} d x d y ; \quad 0 \leq x \leq 2,0 \leq y \leq 1$$
Problem 38

Although it is often true that a double integral can be evaluated by using either $d x$ or $d y$ first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises 37 and 38 in the easiest way possible.
$\iint_{R} 2 x^{3} e^{x^{2} y} d x d y ; \quad 0 \leq x \leq 1,0 \leq y \leq 1$
Problem 39

Evaluate each double integral.
$$\int_{2}^{4} \int_{2}^{x^{2}}\left(x^{2}+y^{2}\right) d y d x$$
Problem 40

Evaluate each double integral.
$$\int_{0}^{4} \int_{0}^{4^{x}} \frac{\sqrt{x y}}{2} d x d y$$
Problem 41

Evaluate each double integral.
$$\int_{0}^{4} \int_{0}^{x} \sqrt{x y} d y d x$$
Matt J.
Georgia Southern University
Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 Problem 18 Problem 19 Problem 20 Problem 21 Problem 22 Problem 23 Problem 24 Problem 25 Problem 26 Problem 27 Problem 28 Problem 29 Problem 30 Problem 31 Problem 32 Problem 33 Problem 34 Problem 35 Problem 36 Problem 37 Problem 38 Problem 39 Problem 40 Problem 41 Problem 42 Problem 43 Problem 44 Problem 45 Problem 46 Problem 47 Problem 48 Problem 49 Problem 50 Problem 51 Problem 52 Problem 53 Problem 54 Problem 55 Problem 56 Problem 57 Problem 58 Problem 59 Problem 60 Problem 61 Problem 62 Problem 63 Problem 64 Problem 65 Problem 66 Problem 67 Problem 68 Problem 69 Problem 70 Problem 71

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Problem 36

Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.
$$z=e^{x+y} ; 0 \leq x \leq 1.0 \leq y \leq 1$$

Answer

$e^{2}-2 e+1$ cubic units.

Chapter 9
Multivariable Calculus
Section 6
Double Integrals
Calculus with Applications

Topics

Double Integrals over Rectangles

Discussion

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Video Transcript

Okay, Another volume problem. So instead of our double integral x and Y are both going from zero to one in a grand A Z to the explosive all high So this is a supersymmetric and a girl I'Ll just put the eggs first of the matter Get on the wife first, Ok? And so anti derivative here. Well, it's just it's stuff. I mean, this is just about as easy as it gets. Justine, evaluate zero to one value extra zero one. So that's just okay. Me too. The one plus why minus c to the y. Okay. And then this is pretty straight forward again. Anti derivatives or just the exponential is themselves. You won. Okay, so if we evaluate at one will get he squared minus a and then minus evaluated zero. Me too. The one minus one. So what is that? He squared minus two e Plus, what

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