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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
XZFXHLO “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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This note was uploaded on 10/02/2011 for the course MATH 1206 taught by Professor Llhanks during the Fall '08 term at Virginia Tech.
 Fall '08
 LLHanks

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