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May 22nd, 2017, 09:53 PM   #1
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Line Integrals

Dear MyMathForum Community:

I need some help with this two exercises. I would appreciate any help.

Thank you.
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 May 23rd, 2017, 03:31 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 I believe that, in English, this would be 1) Let S be the portion of the plane x+ y+ z= 1 in the first quadrant and let C be its boundary oriented in the counter clock wise direction as seen from the point (10, 10, 10). $\vec{F}= (x^3+ e^x, -z^3+ y^2, z^3)$. Find $\int_C \vec{F}\cdot\vec{dr}$. The boundary of that plane in the first quadrant is the triangle with vertices at (1, 0, 0), (0, 1, 0), and (0, 0, 1). That consists of the three line segments (a) x= 1- t, y= t, z= 0 with t going from 0 to 1 (b) x= 0, y= 1- t, z= t with t going from 0 to 1 (c) x= t, y= 0, z= 1- t with t going from 0 to 1. On (a) $\vec{F}= ((1- t)^3+ e^{1- t}, t^2, 0)$ while $\vec{dr}= (-1, 1, 0)dt$. The integral is $\int_0^1 -(1- t)^3- e^{1- t}+ t^2 dt$. The other legs are done similarly. I believe the second problem is (2) Let D be the set ${(x, y)| 1\le x^2+ y^2\le 2, y\ge 0\}$ and let C be the boundary of D oriented positively. If $\vec{F}= \left(-\frac{y^3}{3}, \frac{x^3}{3}\right)$. Calculate $\int_C \vec{F}\cdot \vec{dr}$. Here, C has four parts: (a) the line segment from (1, 0) to (2, 0) (b) the semi-circle from (2, 0) to (-2, 0) (c) the line segment from (-2, 0) to (-1, 0) and (d) the semi-circle from (-1, 0) to (1, 0). The line segment from (1, 0) to (2, 0) can be parameterized as x= t, y= 0 for t from 1 to 2. $\vec{F}= \left(0, \frac{t^3}{3}\right)$ and $\vec{dr}= (1, 0)dt$. The integral is $\int_1^2 0 dt= 0$. The semi- circle from (2, 0) to (-2, 0) can be parameterized as x= 2cos(t), 2 sin(t) for t from 0 to $\pi$. $\vec{F}= \left(-\frac{sin^3(t)}{3}, \frac{cos^3(t)}{3}\right)$. $\vec{dr}= (-2 sin(t), 2 cos(t))dt$. The integral is $\int_0^\pi \left(-\frac{2}{3}sin^4(t)+ \frac{2}{3} cos^4(t)\right)dt$. The other two parts can be done similarly. Thanks from Joppy

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