6.2 Setting Up Integrals: Volume, Density, Average Value Mon Dec 14 Find the area between the following curves

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 =

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 =

Ex 4 Find the total mass M of a 2m rod of linear destiny 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 =

Ex 5 The population in a certain city has radial density function 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 =

Average Value The average value of an integrable function f(x) on [a,b] is the quantity Average value =

Mean Value Theorem If f(x) is continuous on [a,b] then there exists a value c in the interval [a,b] such that

Closure LetFind a value of c in [4,9] such that f(c) is equal to the average of f on [4,9] HW: p.372 #5, 10, 11, 19, 24, 29, 43, 47, 55

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