Question

Math

Posted 2 months ago

```
The following graph shows a Riemann sum:
Which of the following approximates the area between $f(x)$ and the $x$ axis on the interval $[1,11]$ using a midpoint Riemann sum with 5 equal subdivisions?
Choose 1 answer:
(A) $\sum_{i=0}^{4}|5 i+2| \cdot 5$
(B) $\sum_{i=0}^{4}|2 i-3| \cdot 2$
(C) $\sum_{i=0}^{4}|5 i-3| \cdot 5$
() $\sum_{i=0}^{4}|2 i+2| \cdot 2$
```

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 2

The width of each subdivision is $\Delta x = \frac{11 - 1}{5} = 2$

step 3

The midpoints of each subdivision are $x_1 = 2, x_2 = 4, x_3 = 6, x_4 = 8, x_5 = 10$

step 4

The function values at these midpoints are $f(2) = |2 - 5| = 3$, $f(4) = |4 - 5| = 1$, $f(6) = |6 - 5| = 1$, $f(8) = |8 - 5| = 3$, $f(10) = |10 - 5| = 5$

step 5

The Riemann sum is then $2 \cdot (3 + 1 + 1 + 3 + 5) = 2 \cdot 13 = 26$

step 6

Comparing this with the given answer choices, we see that the correct expression is $\sum_{i=0}^{4} |2i - 3| \cdot 2$

B

Key Concept

Midpoint Riemann Sum

Explanation

The midpoint Riemann sum approximates the area under a curve by using the function's value at the midpoint of each subinterval.

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