Find the line integral \int_{C} ( 5x\overrightarrow{i} + (y+9 \overrightarrow{j}) ) \cdot...

Question:

Find the line integral {eq}\int_{C} \left( 5x\overrightarrow{i} + (y+9) \overrightarrow{j} \right) \cdot d\overrightarrow{r} {/eq} where {eq}C {/eq} is the line from {eq}(0, 0) {/eq} to {eq}(0, 5) {/eq}.

Line Integral:

You have to find {eq}\int_{C}\left( 5x\overrightarrow{i} + (y+9) \overrightarrow{j} \right) \cdot d\overrightarrow{r} {/eq} , where C consists of a line segment. Find the vector from the given two points, the line through {eq}\left( {{x_0},{y_0}} \right) {/eq} in the direction of vector {eq}V = \left\langle {p,q} \right\rangle {/eq} has an equation {eq}r\left( t \right) = \left\langle {{x_0} + pt,{y_0} + qt} \right\rangle {/eq}. Find the first derivative and substitute the values in below formula

{eq}\int_C {f\left( {x,y} \right)ds} = \int_a^b {f\left( {x\left( t \right),y\left( t \right)} \right)\sqrt {{{\left( {x'} \right)}^2} + {{\left( {y'} \right)}^2} }} dt. {/eq}

Answer and Explanation:

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From two point P(0,0) and Q(0,5), the vactor will be

{eq}\overrightarrow {PQ} = \left\langle {0,5} \right\rangle - \left\langle {0,0} \right\rangle...

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Line Integrals: How to Integrate Functions Over Paths

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Chapter 15 / Lesson 2
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Many real-world functions are three dimensional, as we live in a 3D world. In this article, you will learn how to integrate 3D functions over general paths through space. This is a basic skill needed for real science and engineering applications.


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