MATH 251 Fall 2023 Section 15 1 Double Integrals over Rectangles In Calculus 1 we nd the area under the curve y f x where a x b by partitioning the interval a b into n subintervals of width x b a n On each subinterval xi 1 xi pick x i Then i 1 f x i x Furthermore the area under the curve is approximated byPn Z b a f x dx lim n 1 f x i x nXi 1 We turn to the volume under the surface z f x y over R x y a x b c y d Divide the rectangle into subrectangles as below where each rectangle has sides x and y 1 2 The area of the rectangle Rij is A x y Hence the area under the surface is approximately mXi 1 nXj 1 f x i y j A We de ne the double integral of f x y over the rectangle R to be x R f x y dA lim m n 1 mXi 1 nXj 1 f x i y j A and we integrate f x y with respect to x Now we talk about how to compute sR Partial integration of f with respect to x The integralR b Partial integration of f with respect to y The integralR d Example 1 FindR 5 and we integrate f x y with respect to y 2 x2 ey dx f x y dA where R is the rectangle a b c d a f x y dx means that y is held xed c f x y dy means that x is held xed Example 2 FindR 1 0 xeydy Integrated intergrals a Z d c Z b Z b a Z d c f x y dxdy Z d f x y dydx Z b c Z b a Z d a c f x y dx dy f x y dy dx Example 3 FindR 1 0 R 2 1 x2 ey dxdy 3 Fubini s Theorem If f is continuous on the rectangle R x y a x b c y d then x R f x y dA Z d c Z b a f x y dxdy Z b a Z d c f x y dydx When f x y g x h y that is if f can be written as the product of a function of x and a function of y then f x y dA Z d c Z b a f x y dxdy Z b a g x dxZ d c h y dy x 0 R 2 Example 4 FindR 1 R 0 y cos x dxdy 4 Example 5 Find sR 3x2 y3 dA where R x y 1 x 2 0 y 3 Example 6 Find sR x 1 x2 dA where R x y 1 x 3 0 y 2 Example 7 Find sR 2xy cos x2y dA where R x y 0 x p 4 0 y 1 5 Theorem If f x y 0 and f is continuous on the rectangle R then sR the solid that lies above R and under the surface z f x y f x y dA gives the volume of Example 8 Find the volume of the solid that is below the surface z 10 x2 y2 and above the rectangle R x y 0 x 1 1 y 3
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