9 Double Integrals Flashcards
How are integrals evaluated?
The Fundamental Theorem of Calculus (FTC) : int(b to a) f(x)dx = F(x)|(b to a) = F(b)-F(a)
What if the definition of a double integral?
The volume under a surface z=f(x,y) or the mass of a plate with variable density p(x,y).
What is dA?
A tiny area element, such as dxdy or dydx in a rectangular problem. Can be thought of the area as the base. (Int)(int)(R) f(x,y) dA = (int)(c to d)(a to b) f(x,y) dxdy
What is the mass of each slice, and how is the total mass found?
The mass of each slide is the density times the area, p(x,y)dA. The total mass is found by using the double integral to sum up all mass slices.
What can be used to find the length of a line segment and area of a rectangle?
Linear density P(x) =1. Area density p(x,y) = 1.
How to set up a Cartesian bounds of integration (x first)?
Left bound, x1 = h1(y) to right bound x2 = h2(y). Y still equals c and d. Then as always, f(x,y)dxdy follows.
How to set up a Cartesian bounds of integration (y first)?
Hold x constant. Y1 = g1(x) and y2 = g2(x)
When does a Cartesian double integral need to be written with multiple integrals?
If one of the bounds change. The formula for D1 is not the same as D2. Some can be done using a single integral if done x first or y first instead.
When should you use polar coordinates?
If there are circles, which use the formula x^2 + y^2 = R^2
How do you write an integral into a polar form (3 steps)?
- Write bounds in term of r and theta 2. Write the function f(x) in terms of r and theta 3. Write the area element, dA = rdrdtheta, in terms of r and theta.
When sliced in polar coordinates, what do the slices look like?
The slices are formed by constant radius r (circles) and lines of constant angle theta (rays). The closer to the origin, the smaller.
What are the four relationships between Cartesian and polar coordinates?
R^2 = x^2 + y^2. X = rcos(theta). Y = rsin(theta). Tan(theta) = y/x
