Double integrals over a rectangular region
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Intros
Lessons
- Double Integrals Over a Rectangular Region Overview:
- Double Integrals Over a Rectangular Region
- Double integral = Volume under f(x,y)
- Iterated Integral
- R=[a,b]×[c,d] is a rectangle
- Integrals are interchangeable
- An Example
- Double Integrals of Products
- Product of two functions
- Move all the x's on one side
- Move all the y's on one side
- Integral both and multiply
- An Example
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Examples
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Topic Notes
Notes:
Double Integrals of Products
Double integrals of a multi-variable function give the volume under the function f(x,y).
Double Integrals Over a Rectangular RegionIf f(x,y) is continuous on a rectangular region R=[a,b]×[c,d], then
∫∫Rf(x,y)dA
gives the volume under the function within that region. The integral is also represented as:
∫∫Rf(x,y)dA=∫ab∫cdf(x,y)dydx
=∫cd∫abf(x,y)dxdy
These integrals are known as iterated integrals.
Double Integrals of Products
If f(x,y)=g(x)h(y) is continuous on a rectangular region R=[a,b]×[c,d], then
∫∫Rf(x,y)dA=∫∫Rg(x)h(y)dA
which can be rewritten as:
∫ab∫cdg(x)h(y)dydx=(∫abg(x)dx)(∫cdh(y)dy)
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