$\begin{aligned} V &= \int_{-r}^{r}A(x) dx = \int_{-r}^{r} \pi(r^2 - x^2) dx \\ &= 2\pi \int_0^r(r^2 - x^2)\ dx \\ &= 2\pi \big[r^2x - \frac{x^3}{3}\big]_0^r = 2\pi(r^3-\frac{r^3}{3}) \\ &= \frac{4}{3}\pi r^3 \end{aligned}$
$A(x) = \pi x^2 - \pi (x^2)^2 = \pi(x^2 - x^4)$
$V = \int_0^1 A(x)dx = \int_0^1 \pi(x^2 - x^4) \ dx = \pi \big[\frac{x^3}{3} - \frac{x^5}{5}\big]_0^1 = \frac{2\pi}{15}$
$\begin{aligned} V &= \int_0^1 2 \pi (2 - x) (x - x^2) dx \\ &= 2 \pi \int_0^1 (x^3-3x^2 + 2x) dx \\ &= 2 \pi \big[\frac{x^4}{4} - x^3 + x^2\big]_0^1 = \frac{\pi}{2} \end{aligned}$
$\begin{aligned} |P_{i-1}P_i| &= \sqrt{(x_i-x_{i-1})^2 + (y_i-y_{i-1})^2} = \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2} \\ &= \Delta x_i \sqrt{1 + (\frac{\Delta y_i}{\Delta x_i})^2} \\ &= \sqrt{1 + (f'(x))^2}dx \end{aligned}$