Q.28 - Calculus

Question

For each function f and interval [a, b] in Exercises 27–33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area. 

$f\left(x\right)=\sin \left(x\right), [a, b] = [0, \pi ]$, n = 3 with

a) Trapezoid sim       b) Upper sum

Step-by-Step Solution

Answer

Using approximation method to approximate the signed area between the graph of f and the x-axis on [a, b] is, 

$$\begin{gathered} a) {\int }_{\hspace{0.17em}0}^{\pi }\sin \left(x\right)dx\approx \frac{\pi }{\sqrt{3}} \\ b){\int }_0^{\pi }\sin \left(x\right)dx=\frac{\pi }{\sqrt{3}} \end{gathered}$$

Part (a) Step 1. Given information.

We have given,

$f\left(x\right)=\sin \left(x\right), [a, b] = [0, \pi ]$ and n = 3 

Part (a) Step 2. Concept used.

The trapezoidal rule uses trapezoids to approximate the area:

$$\begin{gathered} {\int }_a^{b}f\left(x\right)dx\hspace{0.17em}\approx \frac{\Delta x}{2}\left(f\right(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n\left)\right) \\ Where, \\ \Delta \hspace{0.17em}x\hspace{0.17em}=\hspace{0.17em}\frac{b-a}{n} \end{gathered}$$

Where, n is the total number of subintervals.

Upper Riemann sum formula:

${\int }_a^{b}f\left(x\right)dx\hspace{0.17em}\approx \Delta x\left(f\right(x_1)+f(x_2)+f(x_3)+...+f(x_n\left)\right)$

Part (a) Step 3. Explanation.

From the given information in step 1. in part (a), 

$\Delta \hspace{0.17em}x=\frac{\mathrm{\pi }-0}{3}=\frac{\mathrm{\pi }}{3}$

Length of the subintervals is $\frac{\pi }{3}$.

So dividing the interval $[0, \pi ]$ in to the subintervals with length $\frac{\pi }{3}$ is,

$\left[0, \frac{\pi }{3}\right], \left[\frac{\pi }{3}, \frac{2\pi }{3}\right], \left[\frac{2\pi }{3}, \mathrm{\pi }\right]$

So end points are: $0, \frac{\pi }{3}, \frac{2\pi }{3}, \mathrm{\pi }$

Now, just evaluating the function at the left endpoints of the subintervals,

$sin\left(0\right)=0, sin\left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}, sin\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2} \mathrm{and} sin\left(\pi \right)=0$

Using trapezoidal formula,

$$\begin{gathered} {\int }_{\hspace{0.17em}0}^{\pi }\sin \left(x\right)dx\approx \frac{\pi }{6}[0+2\times \frac{\sqrt{3}}{2}+2\times \frac{\sqrt{3}}{2}+0] \\ \approx \frac{\sqrt{3}\pi }{3} \\ \approx \frac{\pi }{\sqrt{3}} \end{gathered}$$

Part (a) Step 4. Conclusion.

Hence, using approximation method to approximate the signed area between the graph of f and the x-axis on [a, b] is, 

${\int }_{\hspace{0.17em}0}^{\pi }\sin \left(x\right)dx\approx \frac{\pi }{\sqrt{3}}$

Part (b) Step 1. Explanation

Using given information from the part (a) in step 1,

$f\left(x\right)=\sin \left(x\right), [a, b] = [0, \mathrm{\pi }]$

Using upper Riemann sum formula,

$\Delta \hspace{0.17em}x=\frac{\mathrm{\pi }-0}{3}=\frac{\mathrm{\pi }}{3}$

So length of the intervals is $\frac{\mathrm{\pi }}{3}$.

Dividing the given interval in to the subintervals with length $\frac{\mathrm{\pi }}{3}$,

So intervals are: $\left[0, \frac{\mathrm{\pi }}{3}\right], \left[\frac{\mathrm{\pi }}{3}, \frac{2\mathrm{\pi }}{3}\right], \left[\frac{2\mathrm{\pi }}{3}, \mathrm{\pi }\right]$

Upper end points are:$\frac{\mathrm{\pi }}{3},\frac{2\mathrm{\pi }}{3},\mathrm{\pi }$

Now just evaluating the functions for those endpoints,

$\mathrm{f}\left(\frac{\mathrm{\pi }}{3}\right)=\frac{\sqrt{3}}{2}, \mathrm{f}\left(\frac{2\mathrm{\pi }}{3}\right)=\frac{\sqrt{3}}{2} \mathrm{and} \mathrm{f}\left(\mathrm{\pi }\right)=0$

Using upper Riemann sum formula,

$$\begin{gathered} {\int }_0^{\pi }\sin \left(x\right)dx\approx \frac{\mathrm{\pi }}{3}\left[\mathrm{f}\left(\frac{\mathrm{\pi }}{3}\right)+ \mathrm{f}\left(\frac{2\mathrm{\pi }}{3}\right)+ \mathrm{f}\left(\mathrm{\pi }\right)\right] \\ \approx \frac{\mathrm{\pi }}{3}\left[\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}+0\right] \\ \approx \frac{\mathrm{\pi }}{\sqrt{3}} \end{gathered}$$

Part (b) Step 2. Conclusion.

Hence, using approximation method to approximate the signed area between the graph of f and the x-axis on [a, b] is, 

${\int }_0^{\pi }\sin \left(x\right)dx=\frac{\pi }{\sqrt{3}}$