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# 2258_07 - Themes for Advanced Placement Calculus 25 7 4.3...

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7 Themes for Advanced Placement Calculus 25 Theme 7 Volumes with Known Cross Sections and Other Applications of Integration Summary Most of the applications of integration are based on the summation interpretation of definite integrals as limits of Riemann Sums. This idea is highlighted on page 412 in the context of finding the area between two curves and for The disk method for finding volumes can be analyzed similarly (page 422). For a horizontal axis of rota- tion, the volume of a representative disk is as indicated in the figure on the right. The concept of work can also be thought of as a summation of increments (page 453) Work 5 o s Force ds Distance d 5 E b a F s x d dx . Volume 5 o s Area of disk ds Thickness d 5 E b a p f R s x dg 2 dx R s x d 2 D x , Area 5 o s Height ds Width d 5 E b a f f s x d 2 g s x dg dx a x b . g s x d , f s x d Key to Text Coverage Section Examples Exercises Topics 4.3 1–6 1–71 Riemann sums, definite integrals and area 6.1 1–5 1–52, 57–81 Area between two curves 6.2 1–7 1–67 Volumes by disks and known cross sections 6.3 1–5 1–43 Volumes by the shell method 6.4 1–7 1–56 Arc length and surfaces of revolution 6.5 1–46 Work x ab x i f g y fx ( ) i ( ) i gx ( ) i ( ) i Height: Width: & & x & x Representative rectangle Rx () & x a b V = π & & [() ] 2 dx

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2258_07 - Themes for Advanced Placement Calculus 25 7 4.3...

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