Calculating Line Integral of C from (1,0) to (3,1)

• popo902
In summary, the conversation involves computing a line integral for a line segment from the point (1,0) to (3,1). The person graphed the line, found its equation, and set up the integral using substitution. However, there was an error in the integral as the (1/2) from dy was multiplied with the dx part as well. Once corrected, the integration was done correctly and the final answer was obtained.
popo902

Homework Statement

Suppose C is the line segment from the point (1,0) to the point (3,1). Compute the line integral
intC {( xdx + (x + y)}dy

The Attempt at a Solution

i graphed the line that connects(1,0) to (1,3) and i got the equation of that line
so y = 1/2(x-1) or y=1/2x -1/2
dy= 1/2
i set x=t
and dx=1
my endpoints of integration became t=1 and t=3
then i plugged everything in
so my integral looked like this
1<= t<= 3 {(t(1) + (t + (1/2t - 1/2)) }1/2

i simplified that to this:
5/4t - 1/4

and i integrated that over 1->3
and i got 9/2...but it's wrong?
can someone tell me what I am doing wrong??

I could fuss about your notation a bit but I won't. In {(t(1) + (t + (1/2t - 1/2)) }1/2 you've got the (1/2) from dy multiplying the dx part too. Make it t+(t+(t/2-1/2))*(1/2).

oh i don't multiply the dy by the whole thing?
so i seprate the dx from the dy
but the points of integration at t are still the same tho right?

I think you want to integrate x*dx+(x+y)*dy. A line integral (x*dx+(x+y))*dy doesn't make much sense. Everything else seems ok.

Last edited:

1. How do you calculate a line integral?

To calculate a line integral, you need to first parameterize the given curve. Then, you can use the formula for line integrals: ∫C F(x,y) ds = ∫a,b F(x(t),y(t)) √(x'(t)^2 + y'(t)^2) dt, where F(x,y) is the function being integrated, C is the curve, and a and b are the start and end points of the curve.

2. What is the difference between a line integral and a regular integral?

A line integral involves integrating a function along a specific curve, while a regular integral involves integrating a function over a specific interval. Line integrals also take into account the direction of the curve, whereas regular integrals do not.

3. What does the starting and ending points of the curve represent in a line integral?

The starting and ending points of the curve represent the interval over which the function is being integrated. In the example given, the starting point is (1,0) and the ending point is (3,1), so the function is being integrated over the interval [1,3].

4. How do you determine the direction of the curve in a line integral?

The direction of the curve is determined by the orientation of the curve and the parametrization used. In the given example, the curve is going from (1,0) to (3,1), so the direction of the curve is from left to right.

5. Can a line integral have a negative value?

Yes, a line integral can have a negative value. This occurs when the direction of the curve and the orientation of the parametrization are opposite, resulting in a negative area under the curve.

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