Section 15 2 Double Integrals over General Regions MATH 251 Fall 2023 We want to integrate f x y not just over rectangles but also over regions of more general shape How to integrate f x y over D cover D with a rectangle R and de ne a function F on R as follows F x y f x y 0 if x y is in D if x y is in R but is not in D x D f x y dA x R F x y dA A plane region D is said to be of type I if it lies between the graphs of two continuous functions of x that is D x y a x b g1 x y g2 x 1 2 If D is of type I then x D f x y dA Z b a Z g2 x g1 x f x y dydx A plane region D is said to be of type II if it lies between the graphs of two continuous functions of y that is D x y c y d h1 y x h2 y If D is of type II then x D f x y dA Z d c Z h2 y h1 y f x y dxdy Example 1 Evaluate sD 2x y dA where D is the region bounded by y 2x2 and y 1 x2 Example 2 Sketch the region D bounded by y x2 y 0 and x 1 Evaluate sD does sD x2 y2 dA measure geometrically 3 x2 y2 dA What Example 3 Sketch the region D bounded by y x2 and y 2x Evaluate sD D as both a type I and type II region x y2 dA by considering 4 Example 4 Set up both a type I and type II integral sD with vertices 0 0 4 0 and 1 3 cos x2y dA where D is the triangular region Example 5 Sketch the region D bounded by the line y x 1 and the parabola y2 2x 6 then set up but do not evalute sD x2 cos xy dA by considering D as both a type I and type II region 5 Example 6 Sketch the region and change the order of integration 6 0 R ex 1 R 1 2 R 2 R sin x 0 0 1 f x y dydx f x y dydx Example 7 EvaluateR 1 0 R 1 x ey2dydx Theorem If f x y 0 and f is continuous on the region D then the volume V of the solid that lies above D and below the surface z f x y is 7 V x D f x y dA Example 8 Find the volume of the solid under the surface z x 2y and above the region bounded by x 1 y 1 and y px 1
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