Natural Science Department Duy Tan University Triple Integrals

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Natural Science Department – Duy Tan University Triple Integrals In this section, we will

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

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

Natural Science Department – Duy Tan University TRIPLE INTEGRALS. Let’s first deal with the simplest case where f is defined on a rectangular box: Triple Integrals s

Natural Science Department – Duy Tan University TRIPLE INTEGRALS The first step is to

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

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 Each sub-box has volume ΔV = Δx Δy Δz Triple Integrals

Natural Science Department – Duy Tan University TRIPLE INTEGRALS Then, we form the triple

Natural Science Department – Duy Tan University TRIPLE INTEGRALS Then, we form the triple Riemann sum where the sample point is in Bijk. Triple Integrals

Natural Science Department – Duy Tan University TRIPLE INTEGRALS The triple integral of f

Natural Science Department – Duy Tan University TRIPLE INTEGRALS The triple integral of f over the box B is: if this limit exists Again, the triple integral always exists if f is continuous. Triple Integrals in Cylindrical Coordinates

Natural Science Department – Duy Tan University TRIPLE INTEGRALS We can choose the sample

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: Triple Integrals

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