Ch13Notes - Ch 13 Lecture Notes Multiple Integrals Section...

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Ch 13 Lecture Notes Multiple Integrals Section 13.1 Double Integrals Generalized Definition from 101A: where is the norm of the partition Ie: the largest of all the . Also the partition P can have irregular subintervals (not all the same width) It is easy to see that the area under the curve f(x) can be calculated from the above definition. 101C Definition: Part of the xy-plane is being partitioned into small rectangles . It is easy to see that the VOLUME under the SURFACE f(x,y) can be calculated from the above definition. Just as we didn’t take a limit to calculate the definite integral, we won’t take a limit to calculate a double integral. Instead we will use what are called iterated integrals: For we first integrate the inside integral with respect to y treating x as a constant. This is referred to as Partial Integration. Once that is done we will be left with our definite integral from 101A (integrating a function of one variable) If our region R is not a rectangular region, then we must use what is known as an Inner Partition to create our subregions . Notice: 1) The outside integral’s limits are constants! Always! 2) The limits on the inner integral are with respect t o the outside variable of integration. 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 .
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You can see from the limits that y goes from to 0 . As we hold y constant, we have the dashed line where you can see x goes from the line : 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 to integrate the inside integral you would get the single integral: This should look familiar. This is how you found area in 101A. Example: Write the area of the unit circle with a definite integral. Now write the area of the unit circle with a double integral. Notice if you integrate the inside integral you end up with the same definite integral right above it. 0 y
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Once you add an integrand f(x,y) , then our double integral represents volume under the surface f(x,y). For example: This double integral represents the volume of a solid. Sketch the solid. represents the surface above the region R (in the xy-plane) given by the limits We just learned the skill of representing the region R from the limits.
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