calculate line integral and draw curve C

**mohammadmurtaza** · December 6th 2009, 07:23 PM

I am truly astonished to the amount of help I received, I'd like to thank Scott, Mr. Fantastic, Chris, and TheEmptySet for all this help you guys have given me. I now feel like I'm no longer alone...

This one is also a tuffy, perhaps someone may want to show me the light?

Question:

Let C be the closed curve C1 + C2 , where C1 is given by

r1 (t) = t i + t^2 j, o <= t <= 1

r2 (t) = (2 - t) i + (2 - t) j , 1 <= t <= 2

a) Draw a sketch of the curve C, with arrows to indicate orientation (im not sure if it's possible to do this online, but a description will be very very much appreciated)

b) Calculate the line integral $\int_C \mathbf{F}\cdot\,dr$ , where F is the vector field: F ( x, y ) = x^2 i + x y j

**Danny** · December 7th 2009, 08:57 AM

Originally Posted by mohammadmurtaza

I am truly astonished to the amount of help I received, I'd like to thank Scott, Mr. Fantastic, Chris, and TheEmptySet for all this help you guys have given me. I now feel like I'm no longer alone...

This one is also a tuffy, perhaps someone may want to show me the light?

Question:

Let C be the closed curve C1 + C2 , where C1 is given by

r1 (t) = t i + t^2 j, o <= t <= 1

r2 (t) = (2 - t) i + (2 - t) j , 1 <= t <= 2

a) Draw a sketch of the curve C, with arrows to indicate orientation (im not sure if it's possible to do this online, but a description will be very very much appreciated)

b) Calculate the line integral $\int_C \mathbf{F}\cdot\,dr$ , where F is the vector field: F ( x, y ) = x^2 i + x y j

$C_1$ is the parabola $y = x^2$ starting at $(0,0)$ and going to $(1,1).$ $C_2$ is a straight line starting at $(1,1)$ and going back to $(0,0)$ . I'll do the first and you the second.

If $x =t$ and $y = t^2$ then $dx = dt$ and $dy = 2t\,dt.$ The line integral becomes

$ \int \limits_{c_1} x^2 dx + xy\,dy = \int_0^1 t^2\, dt + t \cdot t^2 \cdot2t\,dt = \int_0^1 r^2 + 2t^4 \, dt = \frac{11}{15} $

**mohammadmurtaza** · December 8th 2009, 10:26 AM

Thanks Danny, ur the MANN!!! I really appreciate it

However I am stuck on the second one. Since I don't know how to write in the correct notation, can you show me the second one as well?

**Danny** · December 8th 2009, 11:09 AM

Originally Posted by mohammadmurtaza

Thanks Danny, ur the MANN!!! I really appreciate it

However I am stuck on the second one. Since I don't know how to write in the correct notation, can you show me the second one as well?

Originally Posted by Danny

$C_1$ is the parabola $y = x^2$ starting at $(0,0)$ and going to $(1,1).$ $C_2$ is a straight line starting at $(1,1)$ and going back to $(0,0)$ . I'll do the first and you the second.

If $x =t$ and $y = t^2$ then $dx = dt$ and $dy = 2t\,dt.$ The line integral becomes

$ \int \limits_{c_1} x^2 dx + xy\,dy = \int_0^1 t^2\, dt + t \cdot t^2 \cdot2t\,dt = \int_0^1 r^2 + 2t^4 \, dt = \frac{11}{15} $

Here $x = 2 - t, y = 2 - t, t = 1 \to 2$ so $dx = dt, dy=dt$ so the line integral becomes

$ \int_1^2 (2-t)^2 dt + (2-t)(2-t)dt $

although what I might suggest is to follow the line $y = x$ ( $dy=dx$ ) from $x = 1 \to 0$ so

$ \int_1^0 2x^2dx = - \frac{1}{3} $

**mohammadmurtaza** · December 8th 2009, 01:00 PM

Thanks soo much danny!!!

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calculate line integral and draw curve C

Thanks Danny!!!!!

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