6.2 Setting Up Integrals: Volume, Density, Average Value

Mon Dec 14Find 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 functionwhere 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

• Let 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 #5, 10, 11, 19, 24, 29, 43, 47, 55

6.2 Setting up IntegralsMon 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