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

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

II. Physics Applications A. Calculating volume ex: Under surface and over area bounded by y = x and y2 + 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. • Mass • Divide into small rectangles dx, dy on a side • Add up all contrib: y 3 x 4

2) Center of mass From Physics 115:

y R x L M 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

z y R 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,

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

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 3 y 3

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