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6_2 - the Riemann sum V ≈ ∑ n i =1 A x i 4 x This...

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Section 6.2: Volumes Instructor: Ms. Hoa Nguyen ([email protected]) The Volume Problem Find the volume of a solid S that lies between x = a and x = b . The method We start by intersecting S with a plane and obtaining a plane region that is called a cross-section of S . Let A ( x ) be the area of the cross-section of S in a plane P x perpendicular to the x-axis and passing through the point x , where a x b . (Think of slicing S with a knife through x and computing the area of this slice.) The cross-sectional area A ( x ) will vary as x increases from a to b . We divide S into n “slabs” of equal width x using the planes P x 1 , P x 2 , ... to slice the solid. (Think of slicing a loaf of bread.) If we choose sample points x * i in [ x i - 1 , x i ], we can approximate the i th slab S i (the part of S that lies between the planes P x i - 1 and P x i ) by a cylinder with base area A ( x * i ) and height x . The volume of this cylinder is A ( x * i ). So, an approximation to our intuitive conception of the volume of the i th slab S i is: V ( S i ) A ( x * i ) x . Adding the volumes of these slabs, we get an approximation to the total volume (
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Unformatted text preview: the Riemann sum ): V ≈ ∑ n i =1 A ( x * i ) 4 x . This approximation appears to become better and better as n → ∞ . (Think of the slices as becoming thinner and thinner.) Therefore, we define the volume as the limit of these sums as n → ∞ . Also, this limit of Riemann sums is the definite integral : Let S be a solid that lies between x = a and x = b . If the cross-sectional area of S in the plane P x , through x and perpendicular to the x-axis, is A ( x ), where A is a continuous function, then the volume of S is: V = lim n →∞ Σ n i =1 A ( x * i ) 4 x = Z b a A ( x ) dx Example 1 of Section 6.2 (textbook): Example 2 of Section 6.2 (textbook): Example 3 of Section 6.2 (textbook): Example 4 of Section 6.2 (textbook): Example 5 of Section 6.2 (textbook): Example 6 of Section 6.2 (textbook): Example 7 of Section 6.2 (textbook): 1...
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