lecture18-09

# lecture18-09 - 18.02 Multivariable Calculus(Spring 2009...

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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 18 Changing order of integration, double integrals. Applications March 17 Reading Material: From Simmons : 20.2 and 20.3. From Course Notes : I. Last time: Double and iterated integrals in the plane. Today: Changing order of integration, double integrals. Applications 2 Changing order of integration We recall last example from previous lecture: Exercise 1. Consider again the function f ( x, y ) = 20- x 2- y 2 as above. Compute the volume of the solid below the graph of f and above the triangle of vertices (1 , 1) , (2 , 1) and (2 , 3) . Solution: In our resolution we fixed x as the outer variable and its range is [1 , 2], then Inner Integral : Z 2 x- 1 1 (20- x 2- y 2 ) dy = 20 y- x 2 y- 1 3 y 3 2 x- 1 1 = 20(2 x- 1)- x 2 (2 x- 1)- 1 3 (2 x- 1) 3- (20- x 2- 1 3 ) = 2 3 (- 7 x 3 + 9 x 2 + 57 x- 59) = A ( x ) Outer Integral : Z 2 1 A ( x ) dx = 2 3- 7 4 x 4 + 3 x 3 + 57 2 x 2- 59 x 2 = 85 6 . 1 Hence the volume of our solid is V = Z 2 1 Z 2 x- 1 1 (20- x 2- y 2 ) dy dx. Now we want to reverse the order of integration . We draw again the region of integration: We decide to pick y as the outer variable. Then its rangeis [1 , 3]. Now again we use the line3]....
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lecture18-09 - 18.02 Multivariable Calculus(Spring 2009...

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