Question

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=3 x \mathbf{i}+4 y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=t \mathbf{i}+\sqrt{4-t^{2}} \mathbf{j}, \quad-2 \leq t \leq 2\)

Short answer

The final answer is left as an integral expression and you need to compute it. Given the complexity of the integral, it might be more feasible to use a numerical method or some software to solve it.

Step-by-step solution

Calculate \(\mathbf{F}(r(t))\)

Substitute the parametric equations of \(C\), \(x = t\) and \(y = \sqrt{4 - t^{2}}\), into \(\mathbf{F}(x, y)\) to yield \(\mathbf{F}(r(t)) = 3t \mathbf{i} + 4\sqrt{4 - t^{2}} \mathbf{j}\).

Calculate \(d \mathbf{r}\)

Calculate \(d\mathbf{r}\) by taking the derivative of \(r(t)\). That gives \(dr = \mathbf{i} - \frac{t}{\sqrt{4 - t^{2}}} \mathbf{j} dt\).

Find \(\mathbf{F}(r(t)) \cdot d\mathbf{r}\)

Take the dot product of the results of steps 1 and 2 to find \(\mathbf{F}(r(t)) \cdot d\mathbf{r} = 3t - \frac{4t^{2}}{\sqrt{4 - t^{2}}} dt\).

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\)

Evaluate the integral of the dot product from \(t = -2\) to \(t = 2\), \(\int_{-2}^{2} (3t - \frac{4t^{2}}{\sqrt{4 - t^{2}}}) dt\), to get the final result.