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Chap6_Sec2 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

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APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6
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6.2 Volumes APPLICATIONS OF INTEGRATION In this section, we will learn about: Using integration to find out the volume of a solid.
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In trying to find the volume of a solid, we face the same type of problem as in finding areas. VOLUMES
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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
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We start with a simple type of solid called a cylinder or, more precisely, a right cylinder. VOLUMES
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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
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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
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In particular, if the base is a circle with radius r , then the cylinder is a circular cylinder with volume V = πr 2 h. CYLINDERS
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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
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For a solid S that isn’t 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
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We start by intersecting S with a plane and obtaining a plane region that is called a cross-section of S . IRREGULAR SOLIDS
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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. IRREGULAR SOLIDS
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The cross-sectional area A ( x ) will vary as x increases from a to b . IRREGULAR SOLIDS
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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
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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
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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
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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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