Calculus II Chapter 5

solid into thin “slices” whose volumes can be approximated, add the volumes of these “slices” together (a Riemann sum), and finally obtain an exact answer by taking a limit of the sums to get a definite integral. The Building Blocks: Right Solids A right solid is a threedimensional shape swept out by moving a planar region A some distance h along a line perpendicular to the plane of A (Fig. 2). The region A is called a face of the solid, and the word “right” is used to indicate that the movement is along a line perpendicular, at a right angle, to the plane of A. Two parallel cuts produce one slice with two faces (Fig. 3): a slice has volume, and a face has area. Example 1: Suppose there is a fine, uniform mist in the air, and every cubic foot of mist contains 0.02 ounces of water droplets. If you run 50 feet in a straight line through this mist, how wet do you get? Assume that the front (or a cross section) of your body has an area of 8 square feet. Solution: As you run, the front of your body sweeps out a “tunnel” through the mist (Fig. 4). The volume of the tunnel is the area of the front of your body multiplied by the length of the tunnel: volume = (8 ft2 )(50 ft) = 400 ft3 . Since each cubic foot of mist held 0.02 ounces of water which is now on you, you swept out a total of (400 ft3 ). (0.02 oz/ft3 ) = 8 ounces of water. If the water was truly suspended and not falling, would it matter how fast you ran? 5.1 Volumes Contemporary Calculus 2 If A is a rectangle (Fig. 5), then the “right solid” formed by moving A along the line is a 3dimensional solid box B. The volume of B is (area of A). (distance along the line) = (base). (height). (width). If A is a circle with radius r meters (Fig. 6), then the “right solid” formed by moving A along the line h meters is a right circular cylinder with volume equal to {area of A}. {distance along the line} = { _ (r ft)2 }. {h ft} = { _. r 2 ft2 }. { h ft } = _ r 2h ft3 .

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