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Evaluate \int_C (xy) \ dx + (x +y) \ dy along the curve y = 2x^2 from (1,2) to (2,8).

Question:

Evaluate {eq}\int_C (xy) \ dx + (x +y) \ dy {/eq} along the curve {eq}y = 2x^2 {/eq} from (1,2) to (2,8).

Line Integrals:

Suppose that a curve {eq}C {/eq} is defined by {eq}\mathbf r(t)=<x(t),y(t)>,\, a\leq t\leq b {/eq}.

The line integral of {eq}\int_C M(x,y)\, dx+N(x,y)\, dy {/eq} over {eq}C {/eq} is then

{eq}\int_a^b M(x(t),y(t))x'(t)\, dt+N(x(t),y(t))y'(t)\, dt {/eq}.

We also recall that if a curve is defined by {eq}y=f(x),\, a\leq x\leq b {/eq}, then the curve can be written as {eq}\mathbf r(t)=<t,f(t)>,\, a\leq t\leq b {/eq}.

Answer and Explanation:

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Let {eq}C {/eq} be the curve {eq}y=2x^2 {/eq} from (1,2) to (2,8). We can also write this as

{eq}\mathbf r(t)=<t,2t^2>,\, 1\leq t\leq...

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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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