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) $\sum_{i=0}^{4}|5 i-3| \cdot 5$
() $\sum_{i=0}^{4}|2 i+2| \cdot 2$

Question

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) $\sum_{i=0}^{4}|5 i-3| \cdot 5$
() $\sum_{i=0}^{4}|2 i+2| \cdot 2$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The function given is $f(x) = |x - 5|$ and we need to find the midpoint Riemann sum approximation for the interval $[1, 11]$ using 5 equal subdivisions. ‖ ‖ 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.⚹◊What is the formula for calculating the midpoint Riemann sum approximation of an integral using equal subdivisions?⍭ Generate me a similar question◊

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