# Given f(x) cos(2x) and the interval [0, 3 frac{pi}{4}] approximate the area bounded by the graph of f(x) and the axis on the interval using a left, right, and mid point Riemann sum with n = 3

Question
Confidence intervals
Given $$f(x)\ \cos(2x)$$ and the interval
$$[0,\ 3\ \frac{\pi}{4}]$$
approximate the area bounded by the graph of f(x) and the axis on the interval using a left, right, and mid point Riemann sum with $$n = 3$$

2021-01-29

Step 1
Interval $$[0,\ 3\ \frac{\pi}{4}]$$ is to be devided into 3 sub-intervals whos length is given by
$$h=\ \frac{b\ -\ a}{n}=\frac{\frac{3\pi}{4}\ -\ 0}{3}=\frac{\pi}{4}$$
Sep 2
Therefore, the three sub-intervals are
$$\left[0,\ \frac{\pi}{4}\right],\ \left[\frac{\pi}{4},\ \frac{2\pi}{4}\right]\left[\frac{2\pi}{4},\ \frac{3\pi}{4}\right]$$
Step 3
Area bounded by the graph of f(x) and the axis on the interval using a left end point Riemann sum id given by
$$A_{L}=h\left[|f(0)|\ +\ |f\left(\frac{\pi}{4}\right)|\ +\ |f\left(\frac{2\pi}{4}\right)|\right]$$
$$=\frac{\pi}{4}\left[|\cos\ 0|\ +\ |\frac{\cos\ \pi}{2}|\ +\ |\cos\ \pi|\right]=\frac{\pi}{4}[1\ +\ 0\ +\ 1]=\frac{\pi}{2}$$
Step 4
Area bounded by the graph of f(x) and the axis on the interval using a right end point Riemann sum is given by
$$A_{r}=h\left[|f\left(\frac{\pi}{4}\right)|\ +\ |f\left(\frac{2\pi}{4}\right)|\ +\ |f\left(\frac{3\pi}{4}\right)|\right]$$
$$=\frac{\pi}{4}\left[|\cos\ \times\ \frac{\pi}{2}|\ +\ |\cos\ \pi|\ +\ |\cos\ \times\ \frac{3\pi}{2}|\right]=\frac{\pi}{4}[0\ +\ 1\ +\ 0]=\frac{\pi}{4}$$
Step 5
Area bounded by the graph of f(x) and the axis on the interval using a mid point Riemann sum is given by
$$A_{m}=\left[\left|f\left(\frac{\pi}{8}\right)\right|\ +\ \left|f\left(\frac{3\pi}{8}\right)\right|\ +\ \left|f\left(\frac{5\pi}{8}\right)\right|\right]$$
$$=\frac{\pi}{4}\left[\left|\cos\ \times\ \frac{\pi}{4}\right|\ +\ \left|\cos\ \times\ \frac{3\pi}{4}\right|\ +\ \left|\cos\ \times\ \frac{5\pi}{4}\right|\right]=\frac{\pi}{4}\left[\frac{1}{\sqrt{2}}\ +\ \frac{1}{\sqrt{2}}\ +\ \frac{1}{\sqrt{2}}\right]=\frac{3\pi}{4\sqrt{2}}$$

### Relevant Questions

Consider the function $$f(x) = \sin x$$ on th interval [0, 3]. Let P be a uniform partition of [0, 3] with 4 sub-intervals. Compute the left and right Riemann sum of f on the partition. Enter approximate values, rounded to three decimal places.
Consider the following.
$$f(x) = 49 - x^{2}$$
from $$x = 1 to x = 7, 4$$ subintervals
(a) Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals.
(b) Approximate the area under the curve by evaluating the function at the left-hand endpoints of the subintervals.
Consider the function $$f(x) = x^{2}$$ on the interval [1, 9]. Let P be a uniform partition of [1,9] with 16 sub-intervals. Compute the left and right Riemann sum of f on the partition. Use exact values.

The reason ehy the point $$(-1, \frac{3\pi}{2})$$ lies on the polar graph $$r=1+\cos \theta$$ even though it does not satisfy the equation.

Two random variables X and Y with joint density function given by:
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find the marginal density of X.
Two random variables X and Y with joint density function given by:
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find the marginal density of Y.
$$f(x)=5+54x-2x^{3}, [0,4]$$
Suppose that the n-th Riemann sum (with n sub-intervals of equal length), for some function f(x) on the intrval [0, 2], is $$S_{n} = \frac{2}{n^{2}}\ \times\ \sum_{k = 1}^{n}\ k.$$
What is the value of $$\int_{0}^{2}\ f(x)\ dx?$$
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find $$P(x<\frac{1}{3})$$