13.2-13.3

# 13.2-13.3 - 13.2 Double Integrals over General Regions 13.3...

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Unformatted text preview: 13.2 Double Integrals over General Regions 13.3 Area by Double Integrals In R2: In R3: So, in general: Defn. Area The area of a closed, bounded plane region R is w 03» Defn. Volume 55Her The volume of the surface below 2 : f (x, y) and above the region R is Note When setting up integrals, outer limits are ALWAYS co tant I Example 1 Suppose R is bounded by y : m , (x,y) over R. 2 ‘f and (E + y : 6. Find the volume under Hz“ ﬂ XZ-FXHLO “O XT.."l Z Example 2 % ®< k W Evaluate ydA whcrcDis the region bounded byx:2y, x:y, y: 1 9/ \Q \0 D11; TX “KO (badly an”: ' P36“. WM/ 2 Y r X x. \ \f'ni +(I W1 R 5.— oiy. K! 1 Koi‘j 2 QE— :7. 1 1 YZX Zx§1+lgz .a 2X K A1: (R, i. Dix:g(——*~)Ax‘5§'z’; 1/ U V'I ‘ 1K 23‘ J i F ‘1 FL 7. _._‘ 2 2 gym“) ﬂag 9% 2x.) Finding Regions of Integratio '-l 2 ] Examplel A6 3 f(x,y)dydw Example 2 A4/oyf(x,y)dwdy q Example 3 / / ydydw o o Reversing Order of Integration Example 1 //smydydm. 0 z y Example 2 e lnz / / eydydaj. Sketch the region and erse the order of integration. 1 o Example 3 / >< ; m /0 /\/W f(fv,y)dwdy *1 4W “ow 5t..th X": *m i Defn. Average Value of f The average value of f over a re%ion R is 6! \lo “WC _ \ F my.de Example Find the average value of x cos my over the rectangle R: 0 g x 3 7T, 0 g y g l. DO. 1 y2 2. Reverse the order of integration of / / f (:13, y)dxdy. 0 y w»! 5 S Hm) dyotx + \ ﬁg 3. SETUP only. Find the area of the region bounded by x : éyg — 3 and x — y : 1. ‘TUP only. Find the volume of the solid whose base is the region in he cry—plane that is bounded by the parabola y : 4 — x2 and the line y : 33:, while the top of the solid is bounded by the plane z : x + 4. ...
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