Sec2Sess47
Unformatted notes below. Notes on my shorthand method
Session 47: Double Variables
Remember: function of 1 variable int_a_b(f(x))dx = area below graph of f over the range [a,b].
Double integral = volume below graph z=f(x,y) over a region R in the xy-plane. int(int_R(f(x,y)))dA where A is area. Single integral is area, double integral is volume.
Instances of dA add up to create the total area result. Definition: cut R into small pieces of area delta (change) A;
sum_i(f(x_i,y_i)*change_A_i
Adding sections is done by taking the limit as change_A->0 (change in A as it approaches 0) and adding the section. That provides the double integral int(int()).
To compute int(int(f(x,y)dA)_R): take slices. Those slices will combine for the total area.
Let S(x) = area of slice of plane parallel to the yz-plane. The volume = int(S(x)dx) Range is from x_min to x_max of the combined slices creating the integral. For given x, S(x) = int_x_min_to_x_max(f(x,y)dy)
Almost all the time integration is done of R first.
Integrals can be generally thought of as summing the area over a region
Applications of double integrals: 1. Finding the area of region (R). Area(R) = int(int_R(1*dA)) where 1 represents a flat plane in 3d space where volume would occur. In other words, height 1.
Mass (m) of a (flat) object with density (D) = mass per unit area Each piece of the object has change_m = D*change_area
Total mass = int(int_R(D*dA))
Double integrals "scanning" conceptual description: There is an outer integral and an inner integral. For integration of a 3d object the inner integral is similar to scanning each line along the x-axis. Once those scans are processed the outer intergral is moved up one increment in its scan and the process continues until the scan is completed. The intergrals are processed in a way a 3d scan scans along each axis sequentially until the scan is completed.