Question

How do you evaluate the line integral F*Dr along a line segment C from P (-2,4) to Q (-2,11) if F(X,y) = 2xj?

Answer 1

Do you remember how to paramterize a line?
Ex:

\(x=at+x_0\)
\(y=bt+y_0\)

Answer 2

First step: Paramterize a line and create r(t) equation
Second step: Plug the equation you got for x and y in to F(x,y) changing it to F(t)
Third: Take the derivative of r(t), and get dr.
Fourth: See which t numbers will create P (-2,4) to Q (-2,11) in the r(t) equation.

Let us begin with the first step.

Answer 3

To parameterize, you first need to look at the given two points P (-2,4) to Q (-2,11).

\(x=at+x_0\)
a is the final minus initial. P is the initial point, Q is the final point

Answer 4

so \(x=(final ~x- Initital x)t+x_0\)\)

Answer 5

x0 is and y0 can be replaced by either point P or Q

Answer 6

Paramterization is introduced from the last chapter of calculus 2 or the beginning chapter of calculus 3.

Answer 7

x0 and y0 can be replaced by either point P or Q**

Answer 8

@blackstreet23 does it make sense?

Answer 9

I'll parameteriz x

We have two given point Initial =P(-2,4) and Final= Q(-2,11)
The x final is -2 the x initial is -2.
So a= (-2-(-2))
x0 can be either one of the x values of the given points P or Q. I will choose the x value of P which is -2 for my x0 but you will have to do the same for your y0. You have to choose point P which will have a y0 of 4.

So, my x equation is: \(x=(-2(-2))t+(-2)\)
\(x=(-2+2)t-2\)
\(x=(0)t-2\)
\(x=-2\)<-- x equation

Answer 10

Do the same for the equation for y.

Answer 11

and then we are done with the first step.

Answer 12

What is f(t) and r(t) ?

Answer 13

Are you coming tomorrow I am going to sleep ?

Answer 14

-28

Answer 15

ie -4 * 7