# week08 - Math 23 Sections 110-113 B Dodson Week 8 Homework...

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Math 23 Sections 110-113 B. Dodson Week 8 Homework: 15.1 Approximating sums for volume, double integral 15.2 iterated integrals 15.3 general regions 15.4 polar coords Problem 15.3.3: Evaluate the iterated integral Z 3 1 Z 1 0 (1 + 4 xy ) dxdy. Solution: We start with the inside integral: Z 1 0 (1 + 4 xy ) dx and ﬁnd an anti-derivative (under ∂x ) for 1 + 4 xy. Holding y constant, we ﬁnd ∂x ( x + 2 x 2 y ) = 1 + 4 xy, so Z 1 0 (1 + 4 xy ) dx = £ x + 2 x 2 y / 1 0 = (1 + 2 y ) - (0 + 0) = 1 + 2 y. We replace the inside integral with this value, giving the outside integral:

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2 Z 3 1 (1 + 2 y ) dy = £ y + y 2 / 3 1 = (3 + 9) - (1 + 1) = 10 . We recall that this calculates the value of the double integral ZZ R (1 + 4 xy ) dA, over the rectangle R = [0 , 1] × [1 , 3] , deﬁned as the limit of approximating sums that can be regarded as sums of volumes of rectangular solids that approximate the region over R and under the graph of 1 + 4 xy. In particular, our calculation shows that this region has volume 10 (cubic units). Problem 15.1.1a: Use the Riemann sum with
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## This note was uploaded on 02/29/2008 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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week08 - Math 23 Sections 110-113 B Dodson Week 8 Homework...

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