Ch 5 Double and Triple Integrals Review Chapter
Ch. 5 - Double and Triple Integrals > Review Chapter 5: Double and Triple Integrals 1. I. Review 2. Q 1 & Q 2 from preclass
II. Physics Applications A. Calculating volume ex: Under surface *graphs* and over area bounded by y = x and y 2 + x = 2
A. Calculating Volume (continued) ex: same as before 1) Break problem up into little blocks: dx, dy, dz on a side 2) Write down d. V for each block (d. V=dxdydz) 3) Sum up the d. Vs 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. y 3 1) Mass • Divide into small rectangles dx, dy on a side • Add up all contrib: 4 x
2) Center of mass From Physics 115:
C. Moment of Inertia ex: Solid cylinder where mass density ρ=3 x Find I about the center axis. y R x L Moment of inertia of a point mass about an axis 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 y rdθ dx dy θ r x Rectangular Coords Divide into boxes of area d. A=dxdy Polar Coords Divide into pieces of area d. A=dr(rdθ)=rdrdθ
ex: Say we want to calculate the mass of a circular plate Method 1: Rectangular coordinates • d. M=ρdxdy • Slice horiz: y: -3 to 3 x: • So, the total mass is: Method 2: Polar coordinates w/ mass density ρ. 3 z y 3
- Slides: 14