Math 242, Spring 2019, Assignment 9

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Carefully define the following terms, then give one example and one non-example of each:

1. $\int\int_R f(x,y)\,dA$ (the double integral of $f$ over the rectangle $R$).
2. $\int_a^b\int_c^d f(x,y)\,dy\,dx$ (the iterated integral of $f$ with respect to $y$ then $x$).
3. $\int_c^d\int_a^b f(x,y)\,dx\,dy$ (the iterated integral of $f$ with respect to $x$ then $y$.

Carefully state the following theorems (you do not need to prove them):

1. Lagrange multiplier equation for multiple constraints.
2. Fubini's Theorem.

Solve the following problems:

1. Section 14.8, problems 7, 17, 19, 21, and 23.
2. Section 15.1, problems 1, 15, 17, 19, 27, and 33.

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