# 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 function $f\left(x\right)=\mathrm{sin}x$ 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.
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Let f(x) be defined on the closed interval [a,b] and let [a,b] is portioned into n subintervals. Then, the left Riemann sum is defined as $\sum _{i=0}^{n-1}f\left(xi\right)\mathrm{\Delta }x$
where ${x}_{i}$ is the left endpoint of each subinterval and the right Reimann sum is
$\sum _{i=0}^{n}f\left(xi\right)\mathrm{\Delta }x$ where ${x}_{i}$ is the right endpoint of each subinterval.
The function is $\left(x\right)=\mathrm{sin}x$ on the interval [0, 3]. Here the interval is portioned into 4 subintervals. That is, $n=4.$
Find the length of each subinterval $\mathrm{\Delta }$ x as,
$\mathrm{\Delta }x=\frac{b-a}{n}$
$=\frac{3-0}{4}$
$=\frac{3}{4}$
$=0.75$
Therefore the subintervals are, $\left[0,0.75\right],\left[0.75,1.5\right],\left[1.5,2.25\right],\left[2.25,3\right].$
Left sum:
Find the left Reimann sum as follows.
$\sum _{i=0}^{n-1}f\left(xi\right)\mathrm{\Delta }x=0.75\left(f\left(x+0\right)+f\left({x}_{1}\right)+f\left({x}_{2}\right)+f\left({x}_{3}\right)\right)$
$=0.75\left(\mathrm{sin}\left(0\right)+\mathrm{sin}\left(0.75\right)+\mathrm{sin}\left(1.5\right)+\mathrm{sin}\left(2.25\right)\right)$
$\approx 1.843$
Right sum:
Find the Right Reimann sum as follows.
$\sum _{i=1}^{n}f\left({x}_{i}\right)\mathrm{\Delta }x=0.75\left(f\left({x}_{1}\right)+f\left({x}_{2}\right)+f\left({x}_{3}\right)+f\left({x}_{4}\right)\right)$
$=0.75\left(\mathrm{sin}\left(0.75\right)+\mathrm{sin}\left(1.5\right)+\mathrm{sin}\left(2.25\right)+\mathrm{sin}\left(3\right)\right)$
$\approx 1.949$
Left sum: 1.843
Right sum: 1.949