Calc III Ch14 Notes_Part6

Calc III Ch14 Notes_Part6 - If f is integrable over a plane...

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Unformatted text preview: If f is integrable over a plane region R and f(x, y) 2 O for all (x, y) in R, then the volume of the solid region that lies above R and below f is deﬁned as V=j JFK/y dléi NOTE: There are useful properties of double integrals,-but since they paral 31 single integrals so closely, I’m not going to spend much time on them. See text. Fubini’s Theorem is the formal notation given in the text of the methods used to calculate these volumes, here’s my summary: 1. "E ”if, (xtz2 is the integrand jﬂw (If; ﬂ/V/Q w mix/7w; . 2. The base region, R, in the xy—plane is the region on which we’ll use DOM-mt "— fal/ UP ’4” W5 —_____ W- Luci! WW. alxdy afdyolx NOTEzAgain, there are usually 2 ways to set up these double integrals, if you can’t ﬁnd the antiderivatives, try the other way! ! H ! Ex: Find the volume of the solid below f(X, y) = X + 2y and above the region in this; 144 * xy-plane bounded by y = 2x2 and y 2 1+ x2. 1 1 M L {/ﬂ‘x. feBﬂ- {3/— 2/11 f/ “I a f - X’ Z)? 7- +,_ y ...
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