Question

Estimate the area Upper A between the graph of the function f left-parenthesis x right-parenthesis equals 1 0 s i n x and the interval left-bracket 0 comma pi right-bracket Number . Use an approximation scheme with n equals 2 comma 5 and 10 rectangles. Use the right endpoints. If your calculating utility will perform automatic summations, estimate the specified area using n equals 50 and n equals 100 rectangles.

Answer 1

We have f(x) = 10sin(x) and x E [0 , pi]

We are to approximate the area of given curve using the Riemann right end point method.

For n = 2

The width of each rectangle is:

$\Delta x = \frac{b-a}{n} = \frac{\pi - 0}{2} = \frac{\pi}{2}$

Hence the sub intervals are: $[0 , \frac{\pi}{2}] \textup{ , }[\frac{\pi}{2} , \pi]$

For the right hand sum we need to find the sum of the values of the function f(x) at each of the right end points of the above mentioned sub intervals and multiply the sum with the width of each rectangle.

Area using right end point,A₂ = $\Delta x*[f(\frac{\pi}{2}) + f(\pi)]$

                                               = $\frac{\pi}{2}*[10sin(\frac{\pi}{2}) + 10sin(\pi)] = \frac{\pi}{2}*[10*1 + 10*0] = 5\pi = 15.70796$ (approximate value)

For n = 5

The width of each rectangle is:

5

Hence the sub intervals are: $[0 , \frac{\pi}{5}] \textup{ , }[\frac{\pi}{5} , \frac{2\pi}{5}]\textup{ , }[\frac{2\pi}{5} , \frac{3\pi}{5}]\textup{ , }[\frac{3\pi}{5} , \frac{4\pi}{5}]\textup{ , }[\frac{4\pi}{5} , \pi]$

For the right hand sum we need to find the sum of the values of the function f(x) at each of the right end points of the above mentioned sub intervals and multiply the sum with the width of each rectangle.

Area using right end point,A₅ =

4π

                                               = (approximate value)

4π 9 337

For n = 10

The width of each rectangle is:

Ar-10 7n

Hence the sub intervals are: $[0 , \frac{\pi}{10}] \textup{ , }[\frac{\pi}{10} , \frac{\pi}{5}]\textup{ , }[\frac{\pi}{5} , \frac{3\pi}{10}]\textup{ , }[\frac{3\pi}{10} , \frac{2\pi}{5}]\textup{ , }[\frac{2\pi}{5} , \frac{\pi}{2}]\textup{ , }[\frac{\pi}{2} , \frac{3\pi}{5}]\textup{ , }[\frac{3\pi}{5} , \frac{7\pi}{10}]\textup{ , }[\frac{7\pi}{10} , \frac{4\pi}{5}]\textup{ , }[\frac{4\pi}{5} , \frac{9\pi}{10}]\textup{ , }[\frac{9\pi}{10} , \pi]$

For the right hand sum we need to find the sum of the values of the function f(x) at each of the right end points of the above mentioned sub intervals and multiply the sum with the width of each rectangle.

Area using right end point,A₁₀ = $\Delta x*[f(\frac{\pi}{10}) + f(\frac{\pi}{5})+f(\frac{3\pi}{10})+f(\frac{2\pi}{5})+f(\frac{\pi}{2})+f(\frac{3\pi}{5})+f(\frac{7\pi}{10})+f(\frac{4\pi}{5})+f(\frac{9\pi}{10}) + f(\pi)]$

                                               = 19.835235 (approximate value)

                                              

For n = 50

We follow the same procedure as above, here we use the graphing utility

For n = 50

A₅₀ = 19.9934198 (approximate value)

For n = 100

A₁₀₀ = 19.998355 (approximate value)