Question - Analyzing Riemann Sums for Integral Approximation
Solution:
To find the Riemann sum that gives the value of the definite integral $$\int_{-2}^{3} (x^2 + 4x) dx$$, we should identify which sum corresponds to calculating the integral using left-hand, right-hand, or midpoint Riemann sums.The integral given is from -2 to 3. Let's start by defining a partition of the interval [-2, 3] into n subintervals; this means each subinterval has width Δx.Given the boundaries a = -2 and b = 3, the width of each subinterval is:\[ Δx = \frac{{b - a}}{n} = \frac{{3 - (-2)}}{n} = \frac{5}{n} \]The general x-value for the k-th subinterval can be expressed in three different forms depending on the Riemann sum method:- Left-hand: $$ x_k = a + kΔx $$- Right-hand: $$ x_k = a + (k+1)Δx $$- Midpoint: $$ x_k = a + \left(k + \frac{1}{2}\right)Δx $$We need to find out which option corresponds to using one of these forms.(A) The term inside the summation $$ -2 + \left(\frac{3}{2}\frac{k}{n} + \frac{3}{2}\frac{k}{n} \right) $$ doesn't seem to fit any of the standard forms for left, right, or midpoint Riemann sums.(B) $$ -2 + \frac{5k}{n} $$ fits the left-hand rule where $$ x_k = -2 + k Δx $$ because cooresponds $$ Δx = \frac{5}{n} $$.(C) $$ -2 + \frac{5}{2} + \left( \frac{5k}{n} \right) $$ corresponds to the midpoint Riemann sum, where we start at the midpoint of the first interval and add $$ k Δx $$.(D) $$ -2 + \frac{5}{2} \left( \frac{4}{n} + \frac{2}{n}k \right) $$ doesn't match any of the standard forms either.Now using the function $$ f(x) = x^2 + 4x $$, we can express the Riemann sum based on the left-hand rule (which option B represents). The terms inside the summation should be $$ f(x_k)Δx $$, which would be $$ f(-2 + \frac{5k}{n})Δx $$.Substitute $$ f(x) $$ and $$ Δx $$:\[ f(x_k) = (-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n}) \]\[ Δx = \frac{5}{n} \]Hence, the sum is $$ Σ [(-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n})]·\frac{5}{n} $$, which simplifies to:\[ Σ \frac{5}{n}[(-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n})] \]\[ Σ \frac{5}{n}[4 + \frac{25k^2}{n^2} - 20\frac{k}{n} - 8 + 20\frac{k}{n} - \frac{20k}{n}] \]\[ Σ \frac{5}{n}[-4 + \frac{25k^2}{n^2}] \]This expression doesn't match any of the options exactly, but it is closest to option (B), which suggests that there may be an error in the transcription of the options or a simplification error. Thus, option (B) most closely resembles the expression we derived for the Riemann sum based on the left-hand rule:\[ \lim_{n \to \infty} \sum_{k=1}^{n} f(-2 + \frac{5k}{n})·\frac{5}{n} \]However, the terms for $$ f(-2 + \frac{5k}{n}) $$ must be $$ (-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n}) $$ to properly represent the left-hand Riemann sum for this function $$ (x^2 + 4x) $$ over the interval [-2, 3], and these terms should be inside the summation, not outside as shown in option (B). If there is an error in the transcription or presentation of the options, it might be wise to revisit the original problem to ensure that option (B) was copied correctly. Given the provided choices, none match the correct Riemann sum for the integral exactly, but the form of (B) suggests that it's meant to be the left endpoint approximation.
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