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Question
Math
Posted 2 months ago

The following graph shows a Riemann sum:

Which of the following approximates the area between f(x)f(x) and the xx axis on the interval [1,11][1,11] using a midpoint Riemann sum with 5 equal subdivisions?

Choose 1 answer:
(A) i=045i+25\sum_{i=0}^{4}|5 i+2| \cdot 5
(B) i=042i32\sum_{i=0}^{4}|2 i-3| \cdot 2
(C) i=045i35\sum_{i=0}^{4}|5 i-3| \cdot 5
() i=042i+22\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 Δx=1115=2\Delta x = \frac{11 - 1}{5} = 2
step 3
The midpoints of each subdivision are x1=2,x2=4,x3=6,x4=8,x5=10x_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)=25=3f(2) = |2 - 5| = 3, f(4)=45=1f(4) = |4 - 5| = 1, f(6)=65=1f(6) = |6 - 5| = 1, f(8)=85=3f(8) = |8 - 5| = 3, f(10)=105=5f(10) = |10 - 5| = 5
step 5
The Riemann sum is then 2(3+1+1+3+5)=213=262 \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 i=042i32 \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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