Find a function f such that F = nabla f and use this to evaluate integral_C F. dr along the given...

Question:

Find a function {eq}f {/eq} such that {eq}\vec{F} = \vec{\nabla} f {/eq} and use this to evaluate

{eq}\int_C \vec{F}\cdot d\vec{r} {/eq} along the given curve {eq}C {/eq}.

{eq}F (x, y) = x^2\, \vec{i} + y^2 \, \vec{j} . {/eq}

The curve {eq}C {/eq} is the arc of the parabola {eq}y = 2x^2 {/eq} from {eq}(-1,\ 2) {/eq} to {eq}(2,\ 8) {/eq}.

Line Integrals; Fundamental Theorem of Calculus:

The Fundamental Theorem of Calculus for line integrals says:

{eq}\int_{a}^{b}\vec{\nabla} f \cdot d\vec{r}=f(\vec{r}(b))-f(\vec{r}(a)), {/eq}

where {eq}\vec{r}(t) {/eq} is any piece-wise smooth path defined on the interval {eq}[a,b]. {/eq} In particular such line integral is independent of the path chosen as long as it has the same start and endpoint. So if the curve {eq}C {/eq} starts at {eq}A {/eq} and ends at {eq}B {/eq} then:

{eq}\int_C\vec{\nabla} f \cdot d\vec{r}=f(B)-f(A). {/eq}

Answer and Explanation: 1

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This is a line integral along some curve going from {eq}(-1,\ 2) {/eq} to {eq}(2,\ 8) {/eq}. This integral can be evaluated easily if the vector...

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The Fundamental Theorem of Calculus

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Chapter 12 / Lesson 10
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The fundamental theorem of calculus is one of the most important points to understand in mathematics. Learn to define the formula of the fundamental theorem of calculus and explore examples of it put into practice.


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