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imull
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I am really having trouble understanding how upper and lower sums are calculated. In the equations Ʃf(Mi)Δx and Ʃf(mi)Δx, what do Mi and mi represent?
jbunniii said:In other words, the ##M_i## are the heights of the taller (pink + green) rectangles, and the ##m_i## are the heights of the shorter (green only) rectangles.
Oh, weird. You're right. I misread the question to say what it would have said in most books.pwsnafu said:No. ##M_i## are the x values such that ##f(M_i)## are the heights of the taller rectangles. Similarly for ##m_i##.
Upper and lower sums are mathematical concepts used in calculus to approximate the area under a curve. They are calculated by dividing the area under the curve into smaller rectangles and finding the sum of the areas of these rectangles.
To calculate upper and lower sums, you need to first divide the interval of the function into smaller subintervals. Then, find the upper sum by multiplying the length of each subinterval by the maximum value of the function in that interval and adding all the results. Similarly, the lower sum is calculated by multiplying the length of each subinterval by the minimum value of the function in that interval and adding all the results.
Upper and lower sums are used to approximate the area under a curve when the exact value cannot be determined. They are also used to find the Riemann sum, which is a fundamental concept in integral calculus.
By dividing the area under a curve into smaller rectangles, upper and lower sums provide a visual representation of the behavior of a function. They can help in understanding the overall trend of the function and identifying important features such as maximum and minimum values.
Yes, upper and lower sums can be used for any continuous function. However, as the complexity of the function increases, the accuracy of the approximation may decrease. In these cases, advanced numerical methods may be used to calculate the area under the curve.