Q. 3 - Calculus

Question

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.

• the geometric interpretations of the n-rectangle upper sum and lower sum for a function f on an interval $[a, b]$
  

Step-by-Step Solution

Answer

Ans: 

Part (a): The upper sum : ${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f\left(M_i\right)\Delta x$

Part (b): The lower sum :${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f\left(m_i\right)\Delta x$

Step 2. Given information:

The mathematical definition of the n - rectangles upper sum and the lower sum of the function f over an interval $[a, b]$.

Step 2. Defining with a graph and algebraic example :

The total area of the inscribed rectangles is called the lower sum.

The total area of the circumscribed rectangles is called the upper sum.

The upper and lower sums converge to a single value which is the area under the curve.

Let f is a continuous function defined on an interval $[a,b],\mathrm{n}$ is a positive integer.

$\Delta x=\frac{b-a}{n},x_k=a+k\Delta x.$

Then, the upper sum is given by

${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f\left(M_i\right)\Delta x$

Here, $f\left(M_i\right)$ is the maximum value of the f on the subinterval.

The lower sum is given by

${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f\left(m_i\right)\Delta x$

Here, $f\left(m_i\right)$ is the minimum value of the $f$ on the subinterval.