Solve Upper & Lower Sums: Step-by-Step Guide

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In summary, someone please help me with this integral. I have tried for such a long time and have yielded no correct answers.
  • #1
reefster98
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sorry new to this site. Can someone please help me with this? I have tried for such a long time and have yielded no correct answers.

∫(3−5x)dx ======> integral is from 1 to 7

We have n rectangles, so what I did first was found the change in x, which was 6/n which is the width of the rectangles. So Δx= 6/n

I used summation to find the lower sum and upper sum but my answers were wrong.

Someone please help me.

My Lower sum working out:

xi= 1 + iΔx = 1 + 6i/n

To calculate the lower sum, I used the rule Δx\sum_{i=1}^{n}

f(xi) = 3 - 5(1 + 6i/n) = (-30i-2n)/n

substituting it into the sum rule stated above, my answer became 6/n(-17n - 15) = -42 -90/n

This was wrong and I did almost the same for the upper sum too but that too is wrong.

Please help me solve this.
 
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  • #2
Using a little geometry, and the fact that all of the area is below the $x$-axis, we know what we need to find is:

\(\displaystyle I=\int_1^73-5x\,dx=-\left(6\cdot2+\frac{1}{2}\cdot6\cdot30\right)=-102\)

Okay, we now have a value that we can use as a check.

For the left-hand sum, I would state:

\(\displaystyle I_n=\frac{6}{n}\sum_{k=0}^{n-1}\left(3-5\left(1+k\left(\frac{6}{n}\right)\right)\right)\)

\(\displaystyle I_n=-\frac{12}{n^2}\sum_{k=0}^{n-1}\left(n+15k\right)\)

\(\displaystyle I_n=-\frac{12}{n^2}\left(\sum_{k=0}^{n-1}\left(n\right)+15\sum_{k=0}^{n-1}\left(k\right)\right)\)

\(\displaystyle I_n=-\frac{12}{n^2}\left(n^2+15\cdot\frac{n(n-1)}{2}\right)\)

\(\displaystyle I_n=-\frac{6}{n^2}\left(2n^2+15n(n-1)\right)\)

\(\displaystyle I_n=-\frac{6}{n^2}\left(17n^2-15n\right)\)

\(\displaystyle I_n=-\frac{6(17n-15)}{n}\)

Hence:

\(\displaystyle I=\lim_{n\to\infty}I_n=-6\lim_{n\to\infty}\left(17-\frac{15}{n}\right)=-6(17)=-102\)

Can you now do the right-hand sum in a like manner?
 
  • #3
Thank you MarkFL for answering! Your answer and mine were both correct. I just had a syntax typing error (Wondering) haha
 

FAQ: Solve Upper & Lower Sums: Step-by-Step Guide

How do I calculate upper and lower sums?

To calculate upper and lower sums, you first need to divide the interval into smaller subintervals. Then, you need to find the maximum and minimum values of the function on each subinterval. Finally, you multiply the width of each subinterval by the maximum and minimum values and add them together to get the upper and lower sums, respectively.

What is the purpose of upper and lower sums?

Upper and lower sums are used in the Riemann sum method to approximate the area under a curve. They help us to better understand the behavior of a function and can be used to find the exact area under a curve by taking the limit of the Riemann sums as the number of subintervals approaches infinity.

Can upper and lower sums be equal?

Yes, it is possible for upper and lower sums to be equal. This happens when the function is a constant or when it has a special symmetry. In these cases, both the maximum and minimum values on each subinterval will be the same, resulting in equal upper and lower sums.

How do I know if my upper and lower sums are accurate?

The accuracy of upper and lower sums depends on the number of subintervals used. The more subintervals, the closer the upper and lower sums will be to the actual area under the curve. To improve accuracy, you can increase the number of subintervals or use more advanced techniques such as the trapezoidal rule or Simpson's rule.

Can I use upper and lower sums to solve any type of function?

Yes, upper and lower sums can be used to approximate the area under any type of function, as long as the function is defined on the given interval. However, for some functions, such as those with sharp corners or discontinuities, upper and lower sums may not provide an accurate approximation and other methods may be needed.

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