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Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.
Natural Science Department – Duy Tan University Triple Integrals In this section, we will learn about: Triple integrals Lecturer: Ho Xuan Binh Da Nang-02/2015 Natural Science Department – Duy Tan University TRIPLE INTEGRALS Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables Triple Integrals Natural Science Department – Duy Tan University TRIPLE INTEGRALS Let’s first deal with the simplest case where f is defined on a rectangular box: B = { ( x , y , z ) a ≤ x ≤ b, c ≤ y ≤ d , r ≤ z ≤ s } Triple Integrals s Natural Science Department – Duy Tan University TRIPLE INTEGRALS The first step is to divide B into sub-boxes—by dividing: The interval [a, b] into l subintervals [xi-1, xi] of equal width Δx [c, d] into m subintervals of width Δy [r, s] into n subintervals of width Δz Triple Integrals s Natural Science Department – Duy Tan University TRIPLE INTEGRALS The planes through the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub-boxes Bijk = [ xi −1 , xi ] × y j −1 , y j × [ zk −1 , zk ] Each sub-box has volume ΔV = Δx Δy Δz Triple Integrals Natural Science Department – Duy Tan University TRIPLE INTEGRALS Then, we form the triple Riemann l m n sum ∑∑∑ f ( x i =1 j =1 k =1 * ijk where the sample point is in Bijk Triple Integrals * ijk * ijk ) ∆V * ijk * ijk ,y ,z (x * ijk ,y ,z ) Natural Science Department – Duy Tan University TRIPLE INTEGRALS The triple integral of f over the box B is: ∫∫∫ f ( x, y, z ) dV B = lim l , m , n →∞ l m n ∑∑∑ f ( x i =1 j =1 k =1 * ijk * ijk * ijk ,y ,z if this limit exists Again, the triple integral always exists if f is continuous Triple Integrals in Cylindrical Coordinates ) ∆V Natural Science Department – Duy Tan University TRIPLE INTEGRALS We can choose the sample point to be any point in the sub-box However, if we choose it to be the point (xi, yj, zk) we get a simpler-looking expression: ∫∫∫ f ( x, y, z ) dV = B lim l , m , n →∞ l m n ∑∑∑ f ( x , y , z ) ∆V i =1 j =1 k =1 Triple Integrals i j k LOGO Thank you for your attention