Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible - Calculus

Question

Q. 2 The volume of a solid with disk cross-sections whose radii are given by $r\left(x\right)$ on $\left[a, b\right]$, both as a limit of Riemann sums and as a definite integral.

Step-by-Step Solution

Answer

The volume by using definite integral,

$V=\mathrm{\pi }{\int }_a^b{\left[\mathrm{r}\left(\mathrm{x}\right)\right]}^{2}dx$

The volume as a limit of Riemann sums is,

$$\begin{gathered} \mathrm{\pi }{\int }_a^b{\left[\mathrm{r}\left(\mathrm{x}\right)\right]}^{2}dx=\mathrm{\pi }\underset{n\to \infty }{\lim }\sum _{i=1}^{\infty }{\left[\mathrm{r}\left({\mathrm{x}}_{\mathrm{i}}\right)\right]}^2∆x \\ \mathrm{Where}, ∆x=\frac{b-a}{n}, ∆x_i=a+∆x·i \end{gathered}$$

Step 1: Volume of a solid by using definite integral.

If the radius is $r\left(x\right)$

Then, the volume of the solid on the interval $\left[a, b\right]$ is given by a definite integral,

  $V=\mathrm{\pi }{\int }_a^b{\left[\mathrm{r}\left(\mathrm{x}\right)\right]}^{2}dx$

Step 2: Volume of solid by using the limit of Riemann sums.

The definite integral of a continuous function $r\left(x\right)$ over the interval $\left[a, b\right]$denoted by $V=\mathrm{\pi }{\int }_a^b{\left[\mathrm{r}\left(\mathrm{x}\right)\right]}^{2}dx$ , is the limit of a Riemann sum as the number of subdivisions approaches infinity.

Therefore,

$$\begin{gathered} \mathrm{\pi }{\int }_a^b{\left[\mathrm{r}\left(\mathrm{x}\right)\right]}^{2}dx=\mathrm{\pi }\underset{n\to \infty }{\lim }\sum _{i=1}^{\infty }{\left[\mathrm{r}\left({\mathrm{x}}_{\mathrm{i}}\right)\right]}^2∆x \\ \mathrm{Where}, ∆x=\frac{b-a}{n}, ∆x_i=a+∆x·i \end{gathered}$$