Let and compute the Riemann sum of f over the interval , using
Two subintervals of equal length ( ).

Five subintervals of equal length ( ).

Ten subintervals of equal length ( ).

In each case, choose the representative points to be the right end points of the subintervals. Round your answer to two decimal places.

A) 213.75 sq units; 182.47 sq units; 190.08 sq units;
B) 213.75 sq units; 190.08 sq units; 182.47 sq units;
C) 182.47 sq units; 190.08 sq units; 213.75 sq units;
D) 190.08 sq units; 182.47 sq units; 213.75 sq units;

Question

Let  and compute the Riemann sum of f over the interval  , using ​
Two subintervals of equal length (  ).
​
Five subintervals of equal length (  ).
​
Ten subintervals of equal length (  ).
​
In each case, choose the representative points to be the right end points of the subintervals. Round your answer to two decimal places.
​
A) 213.75 sq units; 182.47 sq units; 190.08 sq units; 
B) 213.75 sq units; 190.08 sq units; 182.47 sq units; 
C) 182.47 sq units; 190.08 sq units; 213.75 sq units; 
D) 190.08 sq units; 182.47 sq units; 213.75 sq units;

Quizplus · Accepted Answer

We can use the formula for a right Riemann sum:
${\displaystyle \sum _{i=1}^{n}f(x_{i})\Delta x}$
where $\Delta x$ is the length of each subinterval and $x_i$ is the right endpoint of each subinterval.

For two subintervals of equal length, we have $\Delta x = \frac{4}{2}=2$ and the right endpoints are 9 and 11. Therefore, the Riemann sum is:
$f(9)\Delta x + f(11)\Delta x = f(9)(2) + f(11)(2) = 20\cdot2 + 22\cdot2 = 84$

For five subintervals of equal length, we have $\Delta x = \frac{4}{5}=0.8$ and the right endpoints are 7.6, 8.4, 9.2, 10, and 10.8. Therefore, the Riemann sum is:
$f(7.6)\Delta x + f(8.4)\Delta x + f(9.2)\Delta x + f(10)\Delta x + f(10.8)\Delta x = \
f(7.6)(0.8) + f(8.4)(0.8) + f(9.2)(0.8) + f(10)(0.8) + f(10.8)(0.8) = \
18.24 + 18.88 + 20.96 + 24.64 + 31.36 = 113.08$

For ten subintervals of equal length, we have $\Delta x = \frac{4}{10}=0.4$ and the right endpoints are 7.4, 7.8, 8.2, 8.6, 9, 9.4, 9.8, 10.2, 10.6, and 11. Therefore, the Riemann sum is:
$f(7.4)\Delta x + f(7.8)\Delta x + f(8.2)\Delta x + f(8.6)\Delta x + f(9)\Delta x + \
f(9.4)\Delta x + f(9.8)\Delta x + f(10.2)\Delta x + f(10.6)\Delta x + f(11)\Delta x = \
f(7.4)(0.4) + f(7.8)(0.4) + f(8.2)(0.4) + f(8.6)(0.4) + f(9)(0.4) + \
f(9.4)(0.4) + f(9.8)(0.4) + f(10.2)(0.4) + f(10.6)(0.4) + f(11)(0.4) \
= 17.28 + 17.92 + 18.56 + 19.2 + 19.84 + 20.48 + 21.12 + 21.76 + 22.4 + 23.04 \
= 201.6$

Rounding to two decimal places, we get 84.00, 113.08, and 201.60. Therefore, the best choice is B, which matches these values in the correct order.

Let and Compute the Riemann Sum of F Over