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22-Double Integration

# 22-Double Integration - Double Integrals John E Gilbert...

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Double Integrals John E. Gilbert, Heather Van Ligten, and Benni Goetz The fundamental ideas involved in defining, interpreting and evaluating the integral D f ( x, y ) dxdy of a function z = f ( x, y ) of two variables over a region D in the xy -plane are a lot like the ones for the integral b a f ( x ) dx of a function y = f ( x ) of a function of one variable over an interval [ a, b ] in the x -axis. In one variable the basic idea was that the integral b a f ( x ) dx was the area under the graph of y = f ( x ) on [ a, b ] when f ( x ) 0 . This was made precise by defining the value of the integral as the limit b a f ( x ) dx = lim n → ∞ n k = 1 f ( x * k ) Δ x k of a sum of approximating rectangular areas as shown to the right. Computing the value was done via the Fundamental Theorem of Calculus b a f ( x ) dx = F ( b ) - F ( a ) where F was an anti-derivative of f , i.e. , F ( x ) = f ( x ) . Since the approximating sum made sense whether or not f ( x ) 0 , the limit was then used to define the integral for all f . For a function z = f ( x, y ) of two variables we follow the same route.

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When z = f ( x, y ) is non-negative and D is a region in the xy -plane, the Double Integral D f ( x, y ) dxdy of f over D is the volume of the solid under the graph of f and above D . The volume of some solids can be computed by geometry without using any calculus though the details suggest how we’ll proceed in general. As always, slicing will be the key!!
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22-Double Integration - Double Integrals John E Gilbert...

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