Double integral

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Evaluate $\displaystyle\int_1^3\int_0^2 (y^3-xy)\,dy\,dx$ .

0

8

-4

4

When converting $\displaystyle\iint_R f(x,y)\,dA$ to polar coordinates, $dA$ becomes:

r\,dr\,d\theta

dr\,d\theta

r^2\,dr\,d\theta

\,dr\,d\theta/r

In an iterated integral $\displaystyle\int_a^b\int_c^d f(x,y)\,dy\,dx$ over a rectangle, which statement about the order of integration is correct?

A.Fubini's theorem allows swapping the order of integration when

f

is continuous on the rectangle, giving the same value

B.The order can never be swapped because

dx

and

dy

are not commutative

C.Swapping the order always changes the value of the integral

D.Only polar integrals allow reordering, not Cartesian ones

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