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n7_2 - Section 7.2 Volume courtesy Chia-Rong Chen 1 A...

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Section 7.2 Volume courtesy: Chia-Rong Chen 1. A cylinder has a contant cross section. Its volume is V = Ah where A = area of the base, and h = height. 2. For a solid S with nonuniform cross-sectional area, we use the slicing method : Cut a thin slice perpendicular to the x -axis at x with a thickness dx and a cross-sectional area A ( x ). The approximate volume of this thin slice is then A ( x ) dx . Adding the volumes of all the thin slices and taking the limit of the Riemann sums, we obtain the volume of S as a definite integtal: V = lim bardbl P bardbl→ 0 n summationdisplay i =1 A ( x i x i = integraldisplay b a A ( x ) dx eg.(1) A solid has a right triangular base of 6 units along the x -axis and 2 units along the y -axis. Its cross sections perpendicular to the x -axis are semi-circles. Find its volume. 1
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eg.(2) Find the volume of the solid whose base is the region bounded by the parabola y = x 2 and y = 4. The cross sections perpendicular to the y -axis are equilateral triangles.
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