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Unformatted text preview: Math 241 Chapter 14 Dr. Justin O. Wyss-Gallifent 14.1 Double Integrals 1. These can be defined via a Riemann Sum method like in Calculus I but the net result is: We can define the double integral of f ( x, y ) over R , denoted integraltextintegraltext R f ( x, y ) dA to be the signed volume under the graph of f ( x, y ) within the region R . The question is how to evaluate these things. First... 2. Defn: An iterated integral is a nested integral. An inner integral may have limits of integration which include variables further out. We evaluate these by working from the inside out, making sure we integrate with respect to the correct variable each time. 3. Now then, onto evaluation of integraltextintegraltext R f ( x, y ) dA . (a) R is vertically simple if R may be described as between the two functions y = bot ( x ) and y = top ( x ) on the interval a x b . In this case integraltextintegraltext R f ( x, y ) dA = integraltext b a integraltext top bot f ( x, y ) dy dx . (b) R is horizontally simple if R may be described as between the two functions y = left ( x ) and y = right ( x ) on the interval c y d . In this case integraltextintegraltext R f ( x, y ) dA = integraltext d c integraltext right left f ( x, y ) dx dy . Consider in both cases that if either the top, bottom, left or right function ever changes then you will need more than one integral. 4. We can reparametrize (HS to VS or VS to HS) to do an impossible integral like integraltext 1 integraltext 1 x e ( y 2 ) dy dx . 14.2 Double Integrals in Polar Coordinates 1. Reminder about how polar coordinates work. Shapes well see a lot include things like r = 2, r = 3 cos , r = 2 sin , r = 1 + cos as well as vertical and horizontal lines which need to be converted. Dont forget x = r cos , y = r sin and x 2 + y 2 = r 2 . 2. We describe a region in polar coordinates from the point of view of a person who lives at the origin. There is a near function r = near ( ) and a far function r = far ( ) between two angles . In this case integraltextintegraltext R f ( x, y ) dA = integraltext integraltext far near f ( r cos , r sin ) r dr d . Well see later where that extra...
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- Spring '08