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6.2 Setting Up Integrals: Volume, Density, Average Value Wed March 4 Find the area between the following curves y = x -8, y = x - 2 2 When to use integrals? • Integrals represent quantities that are the “total amount” of something • Area • Volume • Total mass How to set up an integral? • Be able to approximate a quantity by a sum of N terms • Write it as a limit as N approaches infinity • Integrate whatever function determines each Nth term Volume • Lets draw a solid with a base Volume integral • Let A(y) be the area of the horizontal cross section at height y of a solid body extending from y = a to y = b. Then • Volume = ò b a A(y) dy Ex 1 • Calculate the volume V of a pyramid of height 12m whose base is a square of side 4m using an integral Ex 2 • Compute the volume V of the solid whose base is the region between y = 4 – x^2 and the x-axis, and whose vertical cross sections perpendicular to the y-axis are semicircles Ex 3 • Compute the volume of a sphere of radius r using an integral Density and total mass • Consider a rod with length L. If the rod’s mass can be described by a function, then it can also be written as an integral • Total mass M = ò b r (x) dx a Ex 4 • Find the total mass M of a 2m rod of linear destiny r (x) =1+ x(2 - x) kg / m where x is the distance from one end of the rod Population within a radius • Let r be the distance from the center of a city and p(r) be the population density from the center, then • Population P within a radius R = 2p ò R 0 rr (r) dr Ex 5 • The population in a certain city has radial 2 -1/2 density function r (r) =15(1+ r ) where r is the distance from the city center in km and p has units of thousands per square km. How many people live in the ring between 10 and 30km from the city center? Flow rate • Let r = the radius of a tube, and v(r) be the velocity of the particles flowing through the tube, then • Flow rate Q = 2p ò R 0 rv(r) dr Average Value • The average value of an integrable function f(x) on [a,b] is the quantity • Average value = 1 b-a ò b a f (x) dx Mean Value Theorem • If f(x) is continuous on [a,b] then there exists a value c in the interval [a,b] such that 1 f (c) = b-a ò b a f (x) dx Closure • Let f (x) = x Find a value of c in [4,9] such that f(c) is equal to the average of f on [4,9] • HW: p.372 #1-5, 9-17, 39-55 odds 6.2 Setting up Integrals Mon March 9 • Do Now • Find the volume of the solid whose base is the triangle enclosed by x + y = 1, the x-axis, and the y-axis. The cross sections perpendicular to the y-axis are semicircles HW Review Setting up Integrals • If you can break up a measurement into pieces and write it as a sum of a product, then we can use an integral to calculate it. 6.2 Worksheet • Practice Closure • When trying to calculate the volume of a solid, how can we set up the integral given a base and cross-section? 6.2 Review skip • Do Now • The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x + 2y = 8 (a triangle). If cross sections of the solid perpendicular to the xaxis are semicircles, what is the volume of the solid? HW Review p.372 AOO Worksheet Closure • When can we use an integral to measure something? How does it work? • HW: none