Approximate area using partial fractions

[delve]

Determine which value best approximates the area of the region between the x-axis and the graph of $f(x)=\frac{10}{x(x^2+1)}$ over the interval [1,3]. Make your selection on the basis of a sketch of the region and not by performing any calculations. Explain your reasoning.

(a) -6 (b) 6 (c) 3 (d) 5 (e) 8


I am currently studying partial fractions and I must use partial fractions to solve this problem. My book is Calculus, 8th edition, by Larson. The page my problem is on is 560, #56.

At first, I try calculating the range of the area of this graph by calculating the area of the circumscribed and inscribed rectangles, each of which had a width of 1. I got a range of $\frac{1}{3}\leq Area\leq6$. Unfortunately, this still leaves me with possible values of 3, 5, and 6. Any help would be greatly appreciated. Thank you very much!

 

[LCKurtz]

You might try mid-point rectangles instead of upper or lower rectangles. Another possibility is join the points with straight line segments and use trapezoids and see if one of those methods helps. Or maybe both of those involve disallowed "calculations"?

[Edit] Added later -- Try the trapezoids and see if you can argue you are overestimating.

 

[STEMucator]

Determine which value best approximates the area of the region between the x-axis and the graph of $cc$ over the interval [1,3]. Make your selection on the basis of a sketch of the region and not by performing any calculations. Explain your reasoning.

(a) -6 (b) 6 (c) 3 (d) 5 (e) 8I am currently studying partial fractions and I must use partial fractions to solve this problem. My book is Calculus, 8th edition, by Larson. The page my problem is on is 560, #56.

At first, I try calculating the range of the area of this graph by calculating the area of the circumscribed and inscribed rectangles, each of which had a width of 1. I got a range of $\frac{1}{3}\leq Area\leq6$. Unfortunately, this still leaves me with possible values of 3, 5, and 6. Any help would be greatly appreciated. Thank you very much!

If you aren't allowed to integrate I'm not sure how they would expect you to do this.

A rough estimate could be to add $max(f(x)) + min(f(x))$ and average them out. You can easily spot them on a graph if you have one.

Infact, if you want to get more accurate you could take more values on the interval and average them out.

 

[Mark44]

If you aren't allowed to integrate I'm not sure how they would expect you to do this.

There are a number of approximation techniques, as mentioned in LCKurtz's post, such as rectangles, with the height being the function value at the left endpoint, right endpoint, middle, or elsewhere, as well as trapezoids. These are just a few of the numerical methods that can be used to approximate a definite integral.

 

[Mark44]

delve,
Is the goal here to first approximate the integral, and then to calculate the exact value using partial fractions? Your problem statement wasn't clear as to what you need to do.

 

[pasmith]

Determine which value best approximates the area of the region between the x-axis and the graph of $f(x)=\frac{10}{x(x^2+1)}$ over the interval [1,3]. Make your selection on the basis of a sketch of the region and not by performing any calculations. Explain your reasoning.

Look at the graph of f(x) over [1,3]. Do you think that the area under it is clearly more, or less, than that of the triangle with vertices at (1,5), (1,0), and (3,0)? Which of your answers does that eliminate?