Q. 20 - Calculus

Question

Some discontinuous functions are still “nice” enough to be integrable. For example, consider the function

                        $f\left(x\right)=\left\{\begin{array}{l}x, if x\le 2\\ x+3, if x>2\end{array}\right.$

1. Sketch a graph of $f$ on $\left[0,5\right]$. What kind of discontinuity does $f$ have, and where?

2. Why is it reasonable that $f$ is integrable on $\left[0,5\right]$ despite this discontinuity?

3. What happens to a Riemann sum approximation for the area between a function $f$ and the $x$-axis on $\left[0,5\right]$ as $n\to \infty$?

Step-by-Step Solution

Answer

Part $\left(a\right)$: The graph for the function $y=x$ and $y=x+3$ is,

Part $\left(b\right)$: The integration of a function is nothing but the area covered by the graph and the horizontal axis Since, the area itself is defined or finite hence it is integrable.

Part $\left(c\right)$: The Riemann sum will become the limits of integration which is expressed as an equation,

      ${\int }_{-2}^{2}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)\Delta x$.

Part a Step 1 . Given information

$f\left(x\right)=\left\{\begin{array}{l}x, if x\le 2\\ x+3, if x>2\end{array}\right.$

Part a Step 2 . The interval is, 0 , 5 .

The objective is to sketch the graph of the function and then to determine the type of discontinuity the graph has.

The sketch is shown below,

The required portion is shown in blue.

The graph has finite discontinuity at $x=2$.

Part b Step 1 . The objective is to tell the reason to think that the function is not integrable on the interval 0 , 5 .

The integration of a function is nothing but the area covered by the graph and the horizontal axis Since, the area itself is defined or finite hence it is integrable. Therefore, because of the finite area the function is integrable.

Part c Step 1 . Riemann sum approximation for the area between f and the x -axis on 0 , 5 .

As $n\to \infty$, the finite sum $\left(\sum _{}^{}\right)$of $n$ things becomes an integral $\left(\int \right)$ that accumulates everything from $x=0$ to $x=5$. The discrete list of values $f\left(x_k^{*}\right)$ becomes the continuous function $f\left(x\right)$ and the small change $∆x$ becomes an “infinitesimal” change $dx$ which is expressed as an equation,

${\int }_{-2}^{2}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)\Delta x$.