Evaluate the line integral of f(x,y) along the curve C. f(x,y)=\frac{x^4}{\sqrt{1+4y}}, \ C:...
Question:
Evaluate the line integral of f(x,y) along the curve C.
{eq}f(x,y)=\frac{x^4}{\sqrt{1+4y}}, \ C: y=x^2, \ 0\leq x\leq 2\\ a) \ 32\\ b) \ \frac{32}{5}\\ c) \ 8\\ d) \ 0 {/eq}
Line Integrals:
There are many ways to evaluate a line integral, but the initial goal is always the same: we want to reduce it to a regular definite integral. When we have a curve {eq}y = f (x) {/eq}, then we can get everything in terms of {eq}x {/eq} using the following:
{eq}\begin{align*} \int_C g (x,y)\ ds &= \int_a^b g(x, f (x))\ \sqrt{1 + \left( \frac{dy}{dx} \right)^2}\ dx \end{align*} {/eq}
Answer and Explanation:
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View this answerWe expect a nice cancellation here. We have
{eq}\begin{align*} y &= x^2 \\ \frac{dy}{dx} &= 2x \end{align*} {/eq}
And so the differential line...
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Line Integrals: How to Integrate Functions Over Paths
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Chapter 15 / Lesson 2
293
Line integrals are any integral of a function that can be defined along a given curve in a three-dimensional space. Learn the process of line integration and how they can be used to map paths using parametrizations.
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