1. **State the problem:** Evaluate the line integral $$\int_C \mathbf{F} \cdot d\mathbf{r}$$ where $$\mathbf{F} = \cos y \hat{i} + x \sin y \hat{j}$$ and $$C$$ is the curve $$y = \sqrt{1 - x^2}$$ from $$(1,0)$$ to $$(0,1)$$. 2. **Parameterize the curve:** Since $$y = \sqrt{1 - x^2}$$, we can use $$x = t$$ with $$t$$ going from 1 to 0, and $$y = \sqrt{1 - t^2}$$. 3. **Find $$d\mathbf{r}$$:** $$d\mathbf{r} = dx \hat{i} + dy \hat{j} = dt \hat{i} + \frac{d}{dt}(\sqrt{1 - t^2}) dt \hat{j}$$ Calculate $$dy/dt$$: $$\frac{dy}{dt} = \frac{d}{dt} (1 - t^2)^{1/2} = \frac{1}{2}(1 - t^2)^{-1/2} \cdot (-2t) = \frac{-t}{\sqrt{1 - t^2}}$$ So, $$d\mathbf{r} = dt \hat{i} - \frac{t}{\sqrt{1 - t^2}} dt \hat{j}$$ 4. **Evaluate $$\mathbf{F}$$ along the curve:** $$\mathbf{F}(t) = \cos y \hat{i} + x \sin y \hat{j} = \cos(\sqrt{1 - t^2}) \hat{i} + t \sin(\sqrt{1 - t^2}) \hat{j}$$ 5. **Compute the dot product $$\mathbf{F} \cdot d\mathbf{r}$$:** $$\mathbf{F} \cdot d\mathbf{r} = \cos(\sqrt{1 - t^2}) dt + t \sin(\sqrt{1 - t^2}) \left(- \frac{t}{\sqrt{1 - t^2}} dt \right) = \cos(\sqrt{1 - t^2}) dt - \frac{t^2 \sin(\sqrt{1 - t^2})}{\sqrt{1 - t^2}} dt$$ 6. **Set up the integral:** $$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_1^0 \left[ \cos(\sqrt{1 - t^2}) - \frac{t^2 \sin(\sqrt{1 - t^2})}{\sqrt{1 - t^2}} \right] dt$$ 7. **Change the limits to integrate from 0 to 1:** $$= - \int_0^1 \left[ \cos(\sqrt{1 - t^2}) - \frac{t^2 \sin(\sqrt{1 - t^2})}{\sqrt{1 - t^2}} \right] dt$$ 8. **Substitute $$u = \sqrt{1 - t^2}$$:** Then, $$u = \sqrt{1 - t^2} \Rightarrow u^2 = 1 - t^2 \Rightarrow t^2 = 1 - u^2$$ Differentiating, $$du = \frac{-t}{\sqrt{1 - t^2}} dt = -\frac{t}{u} dt \Rightarrow dt = -\frac{u}{t} du$$ 9. **Rewrite the integral in terms of $$u$$:** When $$t=0$$, $$u=1$$; when $$t=1$$, $$u=0$$. The integral becomes: $$- \int_0^1 \left[ \cos u - \frac{(1 - u^2) \sin u}{u} \right] dt$$ Replace $$dt$$: $$dt = -\frac{u}{t} du$$ but since $$t = \sqrt{1 - u^2}$$, $$dt = -\frac{u}{\sqrt{1 - u^2}} du$$ Substitute into the integral: $$- \int_0^1 \left[ \cos u - \frac{(1 - u^2) \sin u}{u} \right] \left(-\frac{u}{\sqrt{1 - u^2}} du \right) = \int_0^1 \left[ \cos u - \frac{(1 - u^2) \sin u}{u} \right] \frac{u}{\sqrt{1 - u^2}} du$$ Simplify inside the integral: $$= \int_0^1 \left[ \frac{u \cos u}{\sqrt{1 - u^2}} - \frac{(1 - u^2) \sin u}{\sqrt{1 - u^2}} \right] du = \int_0^1 \left[ \frac{u \cos u}{\sqrt{1 - u^2}} - \sqrt{1 - u^2} \sin u \right] du$$ 10. **Evaluate the integral:** Split the integral: $$\int_0^1 \frac{u \cos u}{\sqrt{1 - u^2}} du - \int_0^1 \sqrt{1 - u^2} \sin u du$$ 11. **Use integration by parts or numerical methods:** This integral is nontrivial analytically, but the problem likely expects the integral form or numerical approximation. **Final answer:** $$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^1 \left[ \frac{u \cos u}{\sqrt{1 - u^2}} - \sqrt{1 - u^2} \sin u \right] du$$ This is the exact expression for the line integral along the given curve.