Calculus 3

Compute the line integral of the vector field F on a curve $C$, using the parameterization $\mathbf{r}(t)$ from $t = a$ to $t = b$. The line integral is given by $\int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} \, dt$.

Evaluate the integral $\int_{0}^{\frac{\pi}{3}} \sin(x) \cos\left(\frac{\pi}{3} - x\right) \, dx$.

Calculate the volume of a truncated wedge with dimensions: 2 units high, 5 units at the end, 6 units long, and 4 units wide, using a triple integral in rectangular coordinates.

Evaluate the triple integral from 0 to $\pi$ and 0 to 2 and then 0 to $\sqrt{4 - Z^2}$ of $Z \sin Y$ with respect to $dX$ $dZ$ $dy$.

Evaluate the triple integral: $\int_{2}^{4} \int_{-1}^{1} \int_{0}^{3} (y - xz) \, dz \, dy \, dx$.