EXAMPLE 5 Use the Midpoint Rule with n = 5 to approximate the following integral.43x dx2SOLUTION The endpoints of the subintervals are 2, 2.4, 2.8, 3.2, 3.6, and 4, so the midpoints are 2.2, 2.6, 3, 3.4, and . The width of the subintervals is Δx = (4 − 2)/5 = , so the Midpoint Rule gives43x dx2≈ Δx[f(2.2) + f(2.6) + f(3) + f(3.4) + f(3.8)] = 0.432.2 + 32.6 + 33 + 33.4 + 33.8≈ . (Round your answer to four decimal places.)Since f(x) = 3x > 0 for 2 ≤ x ≤ 4, the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in the figure.
Question
EXAMPLE 5 Use the Midpoint Rule with n = 5 to approximate the following integral.43x dx2SOLUTION The endpoints of the subintervals are 2, 2.4, 2.8, 3.2, 3.6, and 4, so the midpoints are 2.2, 2.6, 3, 3.4, and . The width of the subintervals is Δx = (4 − 2)/5 = , so the Midpoint Rule gives43x dx2≈ Δx[f(2.2) + f(2.6) + f(3) + f(3.4) + f(3.8)] = 0.432.2 + 32.6 + 33 + 33.4 + 33.8≈ . (Round your answer to four decimal places.)Since f(x) = 3x > 0 for 2 ≤ x ≤ 4, the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in the figure.
Solution
The text you provided seems to be incomplete and a bit confusing. However, it appears to be a calculus problem asking to use the Midpoint Rule to approximate the integral of 43x from 2 to 4 with n = 5.
Here's how you can solve it:
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First, calculate Δx, which is (b - a)/n. In this case, it's (4 - 2)/5 = 0.4.
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Next, find the midpoints of each subinterval. These are 2.2, 2.6, 3, 3.4, and 3.8.
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Now, substitute these midpoints into the function f(x) = 43x.
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Multiply each result by Δx and sum them up to get the approximation of the integral.
However, there seems to be a mistake in the function. It's written as 43x, but then it's referred to as 3x in the calculations. If the function is 3x, the calculations would look like this:
Δx[f(2.2) + f(2.6) + f(3) + f(3.4) + f(3.8)] = 0.4[32.2 + 32.6 + 33 + 33.4 + 3*3.8]
Calculate the values inside the brackets, sum them up, multiply by 0.4, and round to four decimal places to get the final answer.
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