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2224-Sec13_5-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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Math 2224 Multivariable Calculus–Sec.13.5: Triple Integrals In Rectangular Coordinates I. Review A. Single Integral B. Double Integral II. Triple Integrals A. Triple Integrals are use to calculate the volume of 3-D shapes and the average value of a function over a 3-D region. B. Definition The volume of a closed, bounded region D in space is V = dV D !!! . C. Finding Limits of Integration 1. Sketch the region D along with its “shadow” R (vertical projection) in the xy -plane. Label the bounding surfaces of D and the bounding curves of R . 2. Find the z -limits of integration. These are usually functions of x and y . Lower limit: z = f 1 ( x , y ) , i.e., the bottom surface, upper limit: z = f 2 ( x , y ) , i.e., the top surface 3. Draw Region R in the xy -plane. 4. Find the y –limits of integration. These are usually functions of x . Lower limit: y = g 1 ( x ) , i.e., the bottom curve, upper limit: y = g 2 ( x ) , i.e., the top curve 5. Find the x –limits of integration. These are usually constants. . Lower limit: x=a , i.e., the smallest x -value, upper limit: x=b , i.e., the largest
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