In summary, the Riemann's sum of the <em>quadratic</em> equation f(x) = 2 · x² - 7 represented by the expression A ≈ (1 / 2) · ∑ 2 · [(1 / 4) + i · (1 / 2)]² - (7 / 2) · ∑ 1, for i ∈ {0, 1, 2, 3, 4, 5} and whose picture is the third one. (Correct choice: C) The area of this function is approximately 3 / 8.
The Riemann's sum represents an approximation to the area below the curve by using a <em>finite</em> rectangles of equal length. The more the rectangles, the closer the result to the real area.
<h3>How to find the graph related to the Riemann's sum of the approximate area with midpoints</h3>
Riemann's sums are sum methods that approximates the area "below" the curve and with respect to the <em>horizontal</em> axis.
<em>Approximate</em> area by midpoints includes parts with <em>excess</em> area and parts with <em>truncated</em> area. In the parts where y > 0, each rectangle begins with <em>excess</em> area and finishes with <em>truncated</em> area and where y < 0, each rectangle begins with <em>truncated</em> area and finishes with <em>excess</em> area.
Riemann's sum with midpoints is defined by the following expression:
A ≈ [(b - a) / n] · ∑ f[a + (1 / 2) · [(b - a) / n] + i · [(b - a) / n]], for i ∈ {0, 1, 2, ..., n - 1}
Where:
- a - Lower limit
- b - Upper limit
- n - Number of rectangles with equal width.
- i - Index of the i-th rectangle.
Finally, the equation that represents the approximate area is presented below: (a = 0, b = 3, f(x) = 2 · x² - 7, n = 6)
A ≈ ∑ [(1 / 4) + i · (1 / 2)]² - (7 / 2) · ∑ 1, for i ∈ {0, 1, 2, 3, 4, 5}
The area of the function is approximately:
A ≈ (1 / 4)² + (3 / 4)² + (5 / 4)² + (7 / 4)² + (9 / 4)² + (11 / 4)² - 5 · (7 / 2)
A ≈ 3 / 8
To learn more on Riemann's sum: brainly.com/question/28174121
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