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Unformatted text preview: . Notice: 1) The outside integral’s limits are constants! Always!
the outside variable of integration. 2) The limits on the inner integral are with respect to Examples:
Given Change the order of integration. We first must sketch the region R that is represented by this double integral. The curves that bound the region R come from the limits of the INSIDE integral. We have two
curves: and . Solve these for y and graph both in the xy-plane. and
So we have a line and a circle. Notice that they intersect at and . y
0 You can see from the limits that y goes from
see x goes from the line : to 0 . As we hold y constant, we have the dashed line where you can to the curve . Now we will change the order of integration by noticing that x goes from 0 to
. Holding x constant, we have the red
line. You can see that along the y-axis (we start from the bottom and go up) y starts at the curve then go to the line.
Thus y varies from to . We end up with the new double integral: Realize when the integrand is 1 , the double integral represents the AREA of the bounded region R. If you were t...
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