Quick question on double/triple integrals for area and volume

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The discussion centers on the evaluation of double and triple integrals, specifically addressing the case where the integrand is zero. It is established that if f(x, y) = 0, then the double integral over any area will also yield a result of zero, as demonstrated by the formula ∫∫ 0 dA = ∫∫ 0 r dr dθ = 0. This conclusion is definitive and applies universally to integrals with an identically zero integrand.

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Lifprasir
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I do not know how to formulate formulas on this forum so I just wrote it neatly on a piece of paper and linked it.

http://puu.sh/8fwXr.jpg

Thankss.
 
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Would you edit your post to attach a new image that is smaller? The maximum width should be about 900 pixels. Yours is about 1200 pixels, so doesn't fit in the window.

To answer your question. If f(x, y) = 0, then the integral will be zero.

##\int \int 0 dA = \int \int 0 r dr d\theta = 0##

If the integrand is identically zero for any kind of integral, then the value of the integral is zero.
 

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