Consider the oriented path which is a straight line segment L running from (0,0) to (16, 16 (a) Calculate the line integral of the vector field F = (3x-y) i +j along L using the parameterization B (t) = (2,20, 0 Enter an exact answer. t 8. 256 48 , 48 256). (b) Consider the line integral of the vector field F = (3r-y) i +j along L using the parameterization C(1)-( ,16 3t 32 16$1532 . The line integral calculated in (a) is the line integral of the parameterization given in (b).

This question is missing some parts. Here is the complete question.

Consider the oriented path which is a straight line segment L running from (0,0) to (16,16).

(a) Calculate the line inetrgal of the vector field F = (3x-y)i + xj along line L using the parameterization B(t) = (2t,2t), 0 ≤ t ≤ 8.

Enter an exact answer.

(b) Consider the line integral of the vector field F = (3x-y)i + xj along L using the parameterization C(t) = , 16 ≤ t ≤ 32.

The line integral calculated in (a) is ____________ the line integral of the parameterization given in (b).

(a) 384

              (b) the same as

Step-by-step explanation: Line Integral is the integral of a function along a curve. It has many applications in Engineering and Physics.

It is calculated as the following:

in which (.) is the dot product and r(t) is the given line.

In this question:

(a) F = (3x-y)i + xj

r(t) = B(t) = (2t,2t)

interval [0,8] are the limits of the integral

To calculate the line integral, first substitute the values of x and y for 2t and 2t, respectively or

F(B(t)) = 3(2t)-2ti + 2tj

F(B(t)) = 4ti + 2tj

Second, first derivative of B(t):

B'(t) = (2,2)

Then, dot product between F(B(t)) and B'(t):

F(B(t))·B'(t) = 4t(2) + 8t(2)

F(B(t))·B'(t) = 12t

Now, line integral will be:

Line integral for the conditions in (a) is 384

(b) same function but parameterization is C(t) = :

F(C(t)) =

F(C(t)) =

C'(t) =

Limits are 16 and 32, so line integral will be:

With the same function but different parameterization, line integral is the same.

i believe it is c.y = 9x

rise over run. 9 units up over 1 unit right or 9/1