12 Line Integrals

Question 1

What is a scalar line integral?

Answer 1

Instead of integrating on a straight line segment, it integrates along a curve C, where amount of slide is pads = f(x,y,z)ds

Question 2

How to compute a scalar line integral?

Answer 2

1. Parametrize curve (r(t)).
2. Find bounds from t=a to t=b.
3. Find length of line segment ds. Distance = rate times time. Ds = |r’(T)|dt
4. Rewrite the density function in terms of parametric at ion p = f(x,y,z) = f(x(t),y(t),z(t))
5. Put all together only in terms of t
6. Compute

Question 3

What is ds in scalar line integration?

Answer 3

It is the length of a tiny line element along the curve

Question 4

What does r(t) represent in scalar line integrals?

Answer 4

Position of an object moving along the curve

Question 5

What does r’(t) represent in scalar line integrals?

Answer 5

-the velocity of an object at time t
-a vector that points in the direction tangent to curve

Question 6

what does |r’(t)| represent in a scalar line integral?

Answer 6

The speed the object moves along the curve

Question 7

What is the distance the object moves during the tin amount of time dt?

Answer 7

• (distance) = (Rate) (time)
• ds = |r’(t)| dt

Question 8

in order to parametrize the line segment, what is the vector form of a line?

Answer 8

r(t) = r0 + tv

Question 9

What is a vector field?

Answer 9

Function that outputs a vector F(x,y,z) at every point (x,y,z).

Question 10

What is the gradient of a function?

Answer 10

• Example of a vector field
• ∇f=⟨∂f/∂x,∂f/∂y,∂f/∂z⟩
• Gives the vector that points in the direction of the maximum rate of change and who’s magnitude is the maximum rate of change

Question 11

What is a vector line integral?

Answer 11

It describes the amount of work done by a vector field, F(x,y,z), as an object moves alone the curve/path C
- Work = F ⋅ D

Question 12

If an object moves along curved path C through vector field F(x,y,z), how do you compute the amount of work done along the path?

Answer 12

1. Parametrize curve
2. Slice path into tiny segments, dr (a vector pointing along curve at point r(t), direction of motion) - dr = r’(t)dt
3. Rewrite vector field in terms of parameterization F(r(t) = f(x(t),y(t),z(t))
4. Compute amount of work done by vector field on the tiny segment dW = F ⋅ dr = F(r(t)) ⋅ r’(t)dt
5. Integrate along curve to add all tiny bits of work together to get total work
   - W = (int)C dW = (int) F(x,y,z) ⋅ dr = (int)(a to b) F(r(t)) ⋅r’(t)dt