Finding the Right Riemann Sum Approximation for ∫71f(x)dx using 3 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 find the length 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 find the length of each interval.

The interval [1,7] has a total length of 7 – 1 = 6 units.
Since we want to use 3 intervals of equal length, we divide the total length by 3: 6 / 3 = 2 units per interval.

To calculate the right Riemann sum approximation, we evaluate the function f(x) at the right endpoint of each interval within the given interval [1,7]. The right endpoints of the intervals can be found by adding the interval length to the left endpoint.

The intervals are:
[1, 3]
[3, 5]
[5, 7]

Now, let’s evaluate f(x) at the right endpoints of each interval:

For the interval [1, 3], the right endpoint is 3.
So, f(3) = (3^2 + 1) / √3 + 3 + 5 = 10 / √3 + 3 + 5.

For the interval [3, 5], the right endpoint is 5.
So, f(5) = (5^2 + 1) / √5 + 5 + 5 = 26 / √5 + 5 + 5.

For the interval [5, 7], the right endpoint is 7.
So, f(7) = (7^2 + 1) / √7 + 7 + 5 = 50 / √7 + 7 + 5.

The right Riemann sum approximation is the sum of the function values at the right endpoints of each interval, multiplied by the interval length.

R3 = (10 / √3 + 3 + 5) * 2 + (26 / √5 + 5 + 5) * 2 + (50 / √7 + 7 + 5) * 2.

To simplify further, we can rationalize the denominators:

R3 = (10 / √3 + 3 + 5) * 2 + (26 / √5 + 5 + 5) * 2 + (50 / √7 + 7 + 5) * 2
= (10(√3 – 3 + 5) / (√3 + 3 + 5)) * 2 + (26(√5 – 5 + 5) / (√5 + 5 + 5)) * 2 + (50(√7 – 7 + 5) / (√7 + 7 + 5)) * 2
= (10(√3 + 2) / (√3 + 8)) * 2 + (26√5 / 15) * 2 + (50(√7 – 2) / (√7 + 12)) * 2
= (20(√3 + 2) / (√3 + 8)) + (52√5 / 15) + (100(√7 – 2) / (√7 + 12))

Therefore, the value of the right Riemann sum approximation R3 for ∫71f(x)ⅆx using 3 intervals of equal length is (20(√3 + 2) / (√3 + 8)) + (52√5 / 15) + (100(√7 – 2) / (√7 + 12)).

To determine which of the following statements is true, we need to know the options provided. Please provide the options.

More Answers: