week5 - Math S21a Multivariable calculus Oliver Knill...

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Math S21a: Multivariable calculus Oliver Knill, Summer 2013 Lecture 17: Triple integrals If f ( x, y, z ) is a function of three variables and E is a solid region in space, then integraltext integraltext integraltext E f ( x, y, z ) dxdydz is defined as the n → ∞ limit of the Riemann sum 1 n 3 summationdisplay ( i/n,j/n,k/n ) E f ( i n , j n , k n ) . As in two dimensions, triple integrals can be evaluated by iterated 1D integral computations. Here is a simple example: 1 Assume E is the box [0 , 1] × [0 , 1] × [0 , 1] and f ( x, y, z ) = 24 x 2 y 3 z . integraltext 1 0 integraltext 1 0 integraltext 1 0 24 x 2 y 3 z dz dy dx . To compute the integral we start from the core integraltext 1 0 24 x 2 y 3 z dz = 12 x 3 y 3 , then integrate the middle layer, integraltext 1 0 12 x 3 y 3 dy = 3 x 2 and finally and finally handle the outer layer: integraltext 1 0 3 x 2 dx =1. When we calculate the most inner integral, we fix x and y . The integral is integrating up f ( x, y, z ) along a line intersected with the body. After completing the middle integral, we have computed the integral on the plane z = const intersected with R . The most outer integral sums up all these two dimensional sections. There are two important reductions to compute volume and triple integrals: the ”washer method” and the ”sandwich method”. The washer method from single variable calculus reduces the problem directly to a one dimensional integral. The new sandwich method reduces the problem to a two dimensional integration problem. The washermethod slices the solid along a line. If g ( z ) is the double integral along the two dimensional slice, then integraltext b a g ( z ) dz . The sandwich method sees the solid sandwiched between the graphs of two functions g ( x, y ) and h ( x, y ) over a common two dimensional region R . The integral becomes integraltext integraltext R [ integraltext h ( x,y ) g ( x,y ) f ( x, y, z ) dz ] dA . 2 An important special case of the sandwich method is the volume integraldisplay R integraldisplay f ( x,y ) 0 1 dzdxdy . below the graph of a function f ( x, y ) and above a region R , considered part of the xy plane. It is the integral integraltext integraltext R f ( x, y ) dA . We actually have computed is a triple integral. Think of volume as a triple integral from now on. 1
3 Find the volume of the unit sphere. Solution: The sphere is sandwiched between the graphs of two functions. Let R be the unit disc in the xy plane. If we use the sandwich method , we get V = integraldisplay integraldisplay R [ integraldisplay 1 x 2 y 2 1 x 2 y 2 1 dz ] dA . which gives a double integral integraltext integraltext R 2 1 - x 2 - y 2 dA which is of course best solved in polar coordinates. We have integraltext 2 π 0 integraltext 1 0 1 - r 2 r drdθ = 4 π/ 3. With the washer method which is in this case also called disc method , we slice along the z axes and get a disc of radius 1 - z 2 with area π (1 - z 2 ). This is a method suitable for single variable calculus because we get directly integraltext 1 1 π (1 - z 2 ) dz = 4 π/ 3.

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