Chapter 5: Double and Triple Integrals I. Review Q1 & Q2 from preclass Ch. 5- Double and Triple Integrals > Review

II. Physics Applications A. Calculating volume ex: Under surface and over area bounded by y = x and y 2 + x = 2 *graphs*

A. Calculating Volume (continued) ex: same as before 1) Break problem up into little blocks: dx, dy, dz on a side 2) Write down dV for each block (dV=dxdydz) 3) Sum up the dVs over all the little blocks: Now our only problem is finding the limits in each direction z limits: 0 up to surface x limits: y to y limits: -2 to 1 So,

B. Mass and center of mass ex: Rectangular sheet, mass density Find the mass and center of mass of the sheet. 1)Mass Divide into small rectangles dx, dy on a side Add up all contrib: 3 4 x y

2) Center of mass From Physics 115:

C. Moment of Inertia ex: Solid cylinder where mass density ρ=3x Find I about the center axis. Moment of inertia of a point mass about an axis y L x R M

So, break our cylinder into little pieces dx,dy,dz Then The total moment of inertia is just the sum of the pieces: Limits: x: 0 to L z: -R to R y: So, z y R

III. Other coordinate systems A.2-D: Rectangular vs. polar Rectangular Coords Divide into boxes of area dA=dxdy Polar Coords Divide into pieces of area dA=dr(rd θ)=rdrd θ y dy x dx rd θ θ r

ex: Say we want to calculate the mass of a circular plate w/ mass density ρ. Method 1: Rectangular coordinates dM=ρdxdy Slice horiz: y: -3 to 3 x: So, the total mass is: Method 2: Polar coordinates z y 3 3