Integration

Question 1

What is Fubini’s theorem (2 theorems)?

Answer 1

Let f:R -> R be function of two variables and R = [a,b] x [c,d]. If f is continuous on R, then you can calculate the integral inside and afterwards the integral outside.

If there exists continuous functions of one variable such that f(x,y) = g(x)h(y) forall (x, y) then the integral of f(x, y) can be split up to products between the two above

Question 2

What is a Type I region in R^2 and what is its integral

Answer 2

D is type 1 if there exists continuous functions g1, g2 [a,b] -> R with a <= b such that D is the set of points with x in [a,b] and y in [g1(x), g2(x)]

Integral with g functions inside and x outside (Because of the function F we defined, f can only be inside.)

Question 3

What is a type ii region in R^2 and what is its integral?

Answer 3

If there exists h1, h2 [c,d[ with c <= d such that D is set of all points with x in between [h1, h2] and y in [c,d].

Define F and write f(x, y) inside with functions of x as the first integral to be calculated

Question 4

What is the area of D if it is a region in R^2?

Answer 4

Double integral over D of 1 dA

Question 5

What is a Type I region in R^3 and what is its integral

Answer 5

Let E = susbset containin (x, y, z) where (x,y) are points in D and u1,u2(x,y) form the interval for z. u1,u2 are functions defined on domain of D.

D is the projection of E onto xy-plane, meaning the set of (x,y) such that there exists a z in R with (x,y,z) in E.

Create integral with inner dz and get an expression ito x, y and compute using double integral over general regions.

Question 6

What is a Type II region in R^3 and what is its integral

Answer 6

Let E = susbset containin (x, y, z) where (y,z) are points in D and u1,u2(y, z) form the interval for x. u1,u2 are functions defined on domain of D.

D is the projection of E onto yz-plane, meaning the set of (y,z) such that there exists a z in R with (x,y,z) in E.

Create integral with inner dz and get an expression ito y, z and compute using double integral over general regions.

Question 7

What is a Type III region in R^3 and what is its integral

Answer 7

Let E = susbset containin (x, y, z) where (x, z) are points in D and u1,u2(x, z) form the interval for y. u1,u2 are functions defined on domain of D.

D is the projection of E onto xz-plane, meaning the set of (x,z) such that there exists a z in R with (x,y,z) in E.

Create integral with inner dz and get an expression ito x, z and compute using double integral over general regions.

Question 8

What is the integral of a function f over D if
m <= f(x, y) <= M

Answer 8

m x A(D) <= integral of D <= M x A(D)

Question 9

How do you make a change of variables in double integration?

Answer 9

1. Calculate jacobian (column 1 = u, column 2 = j)
2. Write integral of R = integral of S with f(x = g(u,v), y = h(u,v) )|J|duduv

Question 10

How do you make a change of variables in double integration?

Answer 10

Same as double, but a 3x3 jacobian.
Column 1 = z, column 2 = u, column 3 = v
Remember that determinant calculations are alternating with each iteration