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Double Integrals over Rectangles Slide 2 Let S be the solid bounded below by the region R and above by the graph of f. Goal: Find the volume of S. Divide the region R into sub-rectangles On each rectangle pick a sample point Slide 3 Double Integrals over Rectangles Add this up for all the rectangles to obtain a double Riemann Sum: If we take more sub-divisions having a smaller mesh, we expect this sum to get closer to the exact value of the volume. Slide 4 Double Integrals over Rectangles Example 1 RectangleMidpointf(Midpoint) R 11 (0.5, 1.25)-4.1875 R 12 (0.5, 1.75)-8.6875 R 21 (1.5, 1.25)-3.1875 R 22 (1.5, 1.75)-7.6875 Note that the result is negative since the surface is below the xy plane on the given region R. Slide 5 Double Integrals over Rectangles - Applications If f(x, y) is the population density in a region R, then gives the total population in the region R. If f(x, y) is the mass density (kg/m 2 ) of a lamina shaped like R, then gives the total mass (kg) of the lamina. Slide 6 Double Integrals over Rectangles - Iterated Integrals We want to determine algebraically. First integrate with respect to y (keeping x constant) from y = c to y = d, and then integrate the resulting function of x with respect to x. We can change the order of integration : First integrate with respect to x (keeping y constant) from x = a to x = b and then integrate the resulting function of y with respect to y. Slide 7 Double Integrals over Rectangles Example 2 Evaluate where Method 1: (first integrate w.r.t y, then x ) Method 2: (first integrate w.r.t. x, then y ) Slide 8 Double Integrals over Rectangles Example 3 Find the volume of the solid bounded by the surface and the planes x = 0, x =1, y = 0, y = 1 and z =0. Method 1: (first integrate w.r.t. y ) R