Q. 1 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

The volume of a solid with cross-sectional area function given by $A\left(x\right)$ 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,

$V={\int }_a^{b}A\left(x\right)dx$

The volume as a limit of Riemann sums is,

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

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

If the cross-section area is $A\left(x\right)$

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

$V={\int }_a^{b}A\left(x\right)dx$

 

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

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

Therefore,

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