Rebecca P.

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path from $(0,0,0)$ to $(1,1,1)($ see accompanying figure) given by
$$C_{1} : \quad \mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1$$
$$C_{2} : \quad \mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$
$$C_{3} : \quad \mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$y=2-\frac{1}{2} x, y=0, x=1, x=2 ; \text { about the } x-axis$$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$y=1-x^{2}, y=0 ; \quad \text { about the } x-axis$$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$x=2 \sqrt{y}, x=0, y=9 ; \quad \text { about the } y-axis$$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$y=\ln x, y=1, y=2, x=0 ; \quad \text { about the } y-axis$$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$y=x^{3}, y=x, x \geqslant 0 ; \quad \text { about the } x-axis$$