Estimate the area between f and g for 1 ≤ x ≤ 6.

Question

These are the values from the given table:

x: 0, 1, 2, 3, 4, 5, 6

f(x): 5, 6, 6, 4, 3, 2, 2

g(x): 2, 1, 2, 2, 3, 4, 5

I am not sure if I should apply Riemann sum to this problem or not...

Thank you help in advance!

Answer 1

Hi Viola!

The short answer to your question is yes, you can apply Riemann sums to this problem. However, you'll need to make sure you are careful about your limits. You are given x values in the interval 0 ≤ x ≤ 6, but you only care about the area in the interval 1 ≤ x ≤ 6, so you may not use all the values given. Recall that to use Riemann sums to estimate the area under a curve you can use the formula:

Area ≈ ∑Δx·f(x_i) for i = 1 to n

In this case, our Δx represents the difference between the adjacent x values, so Δx = 1. n can be found by determining how many Δx it takes to span the desired interval, which can be determined by taking the maximum desired interval value (b) and subtracting the minimum desired interval value (a) and dividing the difference by Δx:

n = (b - a)/Δx

In this case, b = 6, a = 1, and Δx = 1, so n = 5, which means we want to use 5 values in the desired interval 1 ≤ x ≤ 6. From this point you can choose to either use the lower Riemann sum using the lower 5 x values (x = 1, 2, 3, 4, 5), or the upper Riemann sum using the upper 5 x values (x = 2, 3, 4, 5, 6).

Now comes the tricky part of this problem. Typically, the area between two curves can be estimated by subtracting the area of the smaller Riemann sum from the area of the larger Riemann sum:

Area_f ≈ ∑Δx·f(x_i) for i = 1 to n

Area_g ≈ ∑Δx·g(x_i) for i = 1 to n

Area_f-g ≈ ∑Δx·(f(x_i)-g(x_i)) for i = 1 to n, assuming f(x) > g(x) for all x

However, in this situation, the two curves cross at x = 4, which creates a situation where the sum would indicate that there are "negative areas" between the curves. In some advanced subjects that "negative area" has a useful meaning, but when considering physical area between curves, it makes no sense. Instead, we want the absolute difference in heights between the two functions, so the equation becomes:

Area_f-g ≈ ∑Δx·|f(x_i)-g(x_i)| for i = 1 to n

The absolute value of the difference between the function values ensures only positive areas. From here, you can plug in the appropriate f(x) and g(x) values and determine your area.

I hope that answers your question!

Seth