2.3.2 Supplementary Notes on Double Integral and Iterated Integral

# 2.3.2 Supplementary Notes on Double Integral and Iterated Integral

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Double Integral and Iterated Integral For a function with only one variable, the definite integral is defined as the area bounded by the curve and the x-axis within a specific region on a two- dimensional plane. A direct extension to bivariate function is given below. () x f b a dx x f x f y = [ b a , ] Definition For a continuous, real-valued bivariate function ( ) y x f , , the volume between the surface ( y x f z , = ) ) and the xy-plane in the region is called the double definite integral of in A , A ( y x f , denoted by ( ) ( ) ∫∫ A y x d y x f , , Note that the volume of the region below the xy-plane ( ( ) 0 , < y x f ) would be counted as negative. If is continuous and well-defined everywhere in A , then the double integral can be evaluated sequentially. For the simple case that A is a rectangle ( y x f , ) [ ] [ ] d c b a , , × , i.e. x ranges from a to b , y ranges from c to d . We can write ∫∫ ∫∫ = b a d c A dydx y x f y x d y x f , , , Note that the inner integral can be evaluated as a univariable integral, with x being treated as a constant. It can be interpreted as the area of the cross-sectional area of the specific

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2.3.2 Supplementary Notes on Double Integral and Iterated Integral

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