Compute ∫C 𝐅 ·d 𝐫 for the oriented curve specified. 𝐅(x, y)=⟨-2, y⟩, half-circle x^2+y^2=1 wit

step 1 For this problem, we are asked to compute the line integral of the vector field F of xY equals t

Compute ∫C 𝐅 ·d 𝐫 for the oriented curve specified. 𝐅(x,...

Compute $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ for the oriented curve specified.
$\mathbf{F}(x, y)=\langle-2, y\rangle, \quad$ half-circle $x^{2}+y^{2}=1$ with $y \geq 0,$ oriented counterclockwise

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Key Concepts

Line Integral of a Vector Field

This concept involves integrating a vector field along a given curve by taking the dot product of the field with the differential element of the curve. It represents the cumulative effect of the vector field along the path, such as the work done by a force field along that path.

Curve Parametrization

Parametrization is the process of expressing the points on a curve using one or more parameters. It simplifies the evaluation of integrals by converting the curve into a function of a parameter, which is especially useful for curves like circles that have natural trigonometric parameterizations.

Orientation of Curves

The orientation of a curve specifies the direction in which the curve is traversed, which directly affects the sign and value of the line integral. In many cases, including this one, the positive or counterclockwise direction is considered standard, influencing how the parameterization is defined.

Dot Product

The dot product is an algebraic operation that multiplies two vectors to produce a scalar, indicating how much one vector extends in the direction of another. In the context of line integrals, the dot product is used to combine the vector field and the derivative of the parametrization, effectively projecting the field along the tangent to the curve.

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