Tutorial 7 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 7 (Solution Notes) 1. Calculate the following iterated integrals: (a) ± b 0 ± a 0 ( x 2 + y 2 ) dxdy (b) ± 2 1 ± 1 0 xy 4 x 2 dxdy . Solution (a) First, regard y as a constant: ± b 0 ± a 0 ( x 2 + y 2 ) dxdy = ± b 0 ² 1 3 x 3 + xy 2 ³ x = a x =0 dy (Regard y as constant) = ± b 0 ´ 1 3 a 3 + ay 2 µ dy = ² 1 3 a 3 y + 1 3 ay 3 ³ b 0 = 1 3 a 3 b + 1 3 ab 3 (b) First, do the inde±nite integration: ± xy 4 x 2 dx = y ± x 4 x 2 dx ( y is a constant here.) = 1 2 y ± 1 4 x 2 dx 2 ( dx 2 =2 xdx ) = 1 2 y ± 1 4 x 2 d (4 x 2 ) (There is a negative sign) = 1 2 y ± u 1 2 du (Substitution) = 1 2 y · 2 u 1 2 · = 1 2 y · ² 2 ¸ 4 x 2 ¹ 1 2 ³ = y · º 4 x 2 Then, ± 2 1 ± 1 0 xy 4 x 2 dxdy = ± 2 1 y · º 4 x 2 · x =1 x =0 dy = ± 2 1 y (3 1 / 2 4 1 / 2 ) dy =( 2 3) ² 1 2 y 2 ³ y =2 y =1 =3 3 2 3 Trick: When we need to use substitution to solve de±nite integral, it is more con- venient to do inde±nite integral instead and substitute the limits later. ± 142
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2. Evaluate the following double integrals: (a) ±± R e x 2 dA , R is the region bounded by y =0 ,y = x, x =1. (b) R ( x + y ) dA , R is the region bounded by the two curves y = x, y = x 2 . Preliminaries Since it is not difficult to deal with iterated integrals, we use the following theorem to change a double integral into an iterated integral. Fubini’s Theorem THEOREM 2 Fubini’s Theorem (Stronger Form) Let ƒ ( x , y ) be continuous on a region R . 1. If R is defined by with and continu- ous on [ a , b ], then 2. If R is defined by with and continuous on [ c , d ], then 6 R ƒ s x , y d dA = L d c L h 2 s y d h 1 s y d ƒ s x , y d dx dy . h 2 h 1 c y d , h 1 s y d x h 2 s y d , 6 R ƒ s x , y d dA = L b a L g 2 s x d g 1 s x d ƒ s x , y d dy dx . g 2 g 1 a x b , g 1 s x d y g 2 s x d , Find the Limits of Integration From the theorem, we know that, when faced with evaluating R f ( x, y ) dA , we only need to fnd the limits oF integration . For example, if the order of the integration is dydx ,thenwe use the following steps: (a) Sketch . Sketch the region of integration and label the bounding curves. (b) Find the y -limits of integration (Two functions of x ) . Imagine a vertical line L cutting through R in the direction of increasing y .Ma rkthe y -values where L enters and leaves. These are the y -limits of integration and are usually functions of x (instead of constants). That is, take a vertical intersection, and fnd the intersection points, where you can fnd the y -limits oF integration. (c) Find the x -limits of integration (Two constant numbers) .C h o o s e x -limits that include all the vertical lines through R . That is, fnd the range oF x in your sketch. 143
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The steps are similar when use the second type of integration dxdy .
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This note was uploaded on 05/05/2009 for the course MA MA1505 taught by Professor Ma1505 during the Spring '09 term at National University of Juridical Sciences.

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Tutorial 7 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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