Calculating the Right Riemann Sum Approximation for ∫71f(x)ⅆx using 3 Equal-Length Intervals

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?

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

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

The total interval is [1, 7], so the difference between the endpoints is:
7 – 1 = 6

To find the width of each interval, we divide the total interval by the number of intervals:
6 ÷ 3 = 2

Now, we will divide the interval [1, 7] into 3 intervals of equal length:
Interval 1: [1, 3]
Interval 2: [3, 5]
Interval 3: [5, 7]

Since we are considering the right Riemann sum, we will use the right endpoint of each interval to evaluate the function f(x).

For Interval 1, the right endpoint is 3. So f(3) = (3^2 + 1)/(√3 + 3 + 5).
For Interval 2, the right endpoint is 5. So f(5) = (5^2 + 1)/(√5 + 5 + 5).
For Interval 3, the right endpoint is 7. So f(7) = (7^2 + 1)/(√7 + 7 + 5).

Next, let’s evaluate these expressions:
f(3) = (9 + 1)/(√3 + 3 + 5) = 10/11.71 (approximately)
f(5) = (25 + 1)/(√5 + 5 + 5) = 26/15.71 (approximately)
f(7) = (49 + 1)/(√7 + 7 + 5) = 50/18.71 (approximately)

Now, we will calculate the right Riemann sum approximation, R3, by multiplying the width of each interval (2) by the corresponding function value and summing them up:

R3 = 2 * f(3) + 2 * f(5) + 2 * f(7)
R3 = 2 * (10/11.71) + 2 * (26/15.71) + 2 * (50/18.71)
R3 = 20/11.71 + 52/15.71 + 100/18.71
R3 = 1.704 + 3.311 + 5.347
R3 ≈ 10.362

Therefore, the right Riemann sum approximation for ∫71f(x)ⅆx using 3 intervals of equal length is approximately 10.362.

The correct statement would be: The value of the right Riemann sum approximation for ∫71f(x)ⅆx using 3 intervals of equal length is approximately 10.362.

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