Calculating Riemann Sum Approximation For Integral Of F(X) Using 3 Intervals: A Math Tutorial

Let f be the function given by f(x)=x2+1x√+x+5. It is known that f is increasing on the interval [1,7]. Let R3 be the value of the right Riemann sum approximation for ∫71f(x)ⅆx using 3 intervals of equal length. Which of the following statements is true?

R3=13.133 and is an overestimate for ∫71f(x)ⅆx∫17f(x)ⅆx.

To calculate the right Riemann sum approximation for ∫71f(x)ⅆx using 3 intervals of equal length, we first need to find the length of each interval.

Length of interval = (b-a)/n where ‘a’ is the lower limit, ‘b’ is the upper limit, and ‘n’ is the number of intervals.

Here, a=1, b=7 and n=3, so the length of each interval is (7-1)/3=2.

The right Riemann sum approximation is given by:
R3 = f(3)Δx + f(5)Δx + f(7)Δx
where Δx is the length of each interval.

Substituting the values, we get:
R3 = f(3)2 + f(5)2 + f(7)2

To find the values of f(3), f(5), and f(7), we substitute the corresponding values of x in the function f(x).

f(3) = (3^2 + 1)/(3^0.5 + 3 + 5) = 10/11.196 = 0.895
f(5) = (5^2 + 1)/(5^0.5 + 5 + 5) = 26/17.748 = 1.466
f(7) = (7^2 + 1)/(7^0.5 + 7 + 5) = 50/22.732 = 2.201

Substituting these values in the formula for R3, we get:
R3 = 0.895*2 + 1.466*2 + 2.201*2 = 9.059

Therefore, the value of the right Riemann sum approximation for ∫71f(x)ⅆx using 3 intervals of equal length is 9.059.

Now, we need to determine which of the following statements is true:

Statement A: R3 underestimates the value of the integral ∫71f(x)ⅆx
Statement B: R3 overestimates the value of the integral ∫71f(x)ⅆx
Statement C: R3 is an exact value for the integral ∫71f(x)ⅆx

To determine whether R3 underestimates or overestimates the value of the integral, we need to calculate the actual value of the integral.

∫71f(x)ⅆx = [2/3(x^1.5) + 2ln(x+x^0.5)] from 1 to 7
∫71f(x)ⅆx = [2/3(7^1.5) + 2ln(7+7^0.5)] – [2/3(1^1.5) + 2ln(1+1^0.5)]
∫71f(x)ⅆx = 26.695 – 1.178 = 25.517

Comparing the actual value of the integral with the value of R3, we can see that R3 overestimates the actual value of the integral (∫71f(x)ⅆx > 9.059).

Therefore, the correct statement is:

Statement B: R3 overestimates the value of the integral ∫71f(x)ⅆx.

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