Answer:
The right Riemann sum is 21.5.
The left Riemann sum is 29.5.
Step-by-step explanation:
The right Riemann sum (also known as the right endpoint approximation) uses the right endpoints of a sub-interval:
, where
.
To find the Riemann sum for
with 4 rectangles, using right endpoints you must:
We have that a = 0, b = 2, n = 4. Therefore,
.
Divide the interval [0,2] into n = 4 sub-intervals of length
:
![\left[0, \frac{1}{2}\right], \left[\frac{1}{2}, 1\right], \left[1, \frac{3}{2}\right], \left[\frac{3}{2}, 2\right]](https://tex.z-dn.net/?f=%5Cleft%5B0%2C%20%5Cfrac%7B1%7D%7B2%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%2C%201%5Cright%5D%2C%20%5Cleft%5B1%2C%20%5Cfrac%7B3%7D%7B2%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B3%7D%7B2%7D%2C%202%5Cright%5D)
Now, we just evaluate the function at the right endpoints:
![f\left(x_{1}\right)=f\left(\frac{1}{2}\right)=\frac{33}{2}=16.5\\\\f\left(x_{2}\right)=f\left(1\right)=12\\\\f\left(x_{3}\right)=f\left(\frac{3}{2}\right)=\frac{17}{2}=8.5\\\\f\left(x_{4}\right)=f(b)=f\left(2\right)=6](https://tex.z-dn.net/?f=f%5Cleft%28x_%7B1%7D%5Cright%29%3Df%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%3D%5Cfrac%7B33%7D%7B2%7D%3D16.5%5C%5C%5C%5Cf%5Cleft%28x_%7B2%7D%5Cright%29%3Df%5Cleft%281%5Cright%29%3D12%5C%5C%5C%5Cf%5Cleft%28x_%7B3%7D%5Cright%29%3Df%5Cleft%28%5Cfrac%7B3%7D%7B2%7D%5Cright%29%3D%5Cfrac%7B17%7D%7B2%7D%3D8.5%5C%5C%5C%5Cf%5Cleft%28x_%7B4%7D%5Cright%29%3Df%28b%29%3Df%5Cleft%282%5Cright%29%3D6)
Finally, just sum up the above values and multiply by
:
![\frac{1}{2}(16.5+12+8.5+6)=21.5](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%2816.5%2B12%2B8.5%2B6%29%3D21.5)
The left Riemann sum (also known as the left endpoint approximation) uses the left endpoints of a sub-interval:
, where
.
To find the Riemann sum for
with 4 rectangles, using left endpoints you must:
We have that a = 0, b = 2, n = 4. Therefore,
.
Divide the interval [0,2] into n = 4 sub-intervals of length
:
![\left[0, \frac{1}{2}\right], \left[\frac{1}{2}, 1\right], \left[1, \frac{3}{2}\right], \left[\frac{3}{2}, 2\right]](https://tex.z-dn.net/?f=%5Cleft%5B0%2C%20%5Cfrac%7B1%7D%7B2%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%2C%201%5Cright%5D%2C%20%5Cleft%5B1%2C%20%5Cfrac%7B3%7D%7B2%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B3%7D%7B2%7D%2C%202%5Cright%5D)
Now, we just evaluate the function at the left endpoints:
![f\left(x_{0}\right)=f(a)=f\left(0\right)=22\\\\f\left(x_{1}\right)=f\left(\frac{1}{2}\right)=\frac{33}{2}=16.5\\\\f\left(x_{2}\right)=f\left(1\\\right)=12\\\\f\left(x_{3}\right)=f\left(\frac{3}{2}\right)=\frac{17}{2}=8.5](https://tex.z-dn.net/?f=f%5Cleft%28x_%7B0%7D%5Cright%29%3Df%28a%29%3Df%5Cleft%280%5Cright%29%3D22%5C%5C%5C%5Cf%5Cleft%28x_%7B1%7D%5Cright%29%3Df%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%3D%5Cfrac%7B33%7D%7B2%7D%3D16.5%5C%5C%5C%5Cf%5Cleft%28x_%7B2%7D%5Cright%29%3Df%5Cleft%281%5C%5C%5Cright%29%3D12%5C%5C%5C%5Cf%5Cleft%28x_%7B3%7D%5Cright%29%3Df%5Cleft%28%5Cfrac%7B3%7D%7B2%7D%5Cright%29%3D%5Cfrac%7B17%7D%7B2%7D%3D8.5)
Finally, just sum up the above values and multiply by
:
![\frac{1}{2}(22+16.5+12+8.5)=29.5](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%2822%2B16.5%2B12%2B8.5%29%3D29.5)