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Unformatted text preview: APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6 6.2 Volumes APPLICATIONS OF INTEGRATION In this section, we will learn about: Using integration to find out the volume of a solid. In trying to find the volume of a solid, we face the same type of problem as in finding areas. VOLUMES We have an intuitive idea of what volume means. However, we must make this idea precise by using calculus to give an exact definition of volume. VOLUMES We start with a simple type of solid called a cylinder or, more precisely, a right cylinder. VOLUMES As illustrated, a cylinder is bounded by a plane region B 1 , called the base, and a congruent region B 2 in a parallel plane. The cylinder consists of all points on line segments perpendicular to the base and join B 1 to B 2 . CYLINDERS If the area of the base is A and the height of the cylinder (the distance from B 1 to B 2 ) is h , then the volume V of the cylinder is defined as: V = Ah CYLINDERS In particular, if the base is a circle with radius r , then the cylinder is a circular cylinder with volume V = r 2 h. CYLINDERS If the base is a rectangle with length l and width w , then the cylinder is a rectangular box (also called a rectangular parallelepiped) with volume V = l wh. RECTANGULAR PARALLELEPIPEDS For a solid S that isnt a cylinder, we first cut S into pieces and approximate each piece by a cylinder. We estimate the volume of S by adding the volumes of the cylinders. We arrive at the exact volume of S through a limiting process in which the number of pieces becomes large. IRREGULAR SOLIDS We start by intersecting S with a plane and obtaining a plane region that is called a crosssection of S . IRREGULAR SOLIDS Let A ( x ) be the area of the crosssection of S in a plane P x perpendicular to the xaxis 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. IRREGULAR SOLIDS The crosssectional area A ( x ) will vary as x increases from a to b . IRREGULAR SOLIDS 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. IRREGULAR SOLIDS 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 and ) by a cylinder with base area A ( x i * ) and height x . 1 i x P i x P IRREGULAR SOLIDS 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: ( ) ( *) i V S A xi x IRREGULAR SOLIDS Adding the volumes of these slabs, we get an approximation to the total volume (that is, what we think of intuitively as the volume): This approximation appears to become better and better as n ....
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.
 Fall '11
 AlanS.Grave

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