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 evaluate the right Riemann sum approximation, we need to divide the interval [1, 7] into equal subintervals and choose sample points within each subinterval
To evaluate the right Riemann sum approximation, we need to divide the interval [1, 7] into equal subintervals and choose sample points within each subinterval.
Since we are using 3 intervals of equal length, we divide the interval [1, 7] into three subintervals: [1, 3], [3, 5], and [5, 7]. The length of each subinterval is (7-1)/3 = 6/3 = 2.
For the right Riemann sum approximation, we take the right endpoint of each subinterval as the sample point.
In the first subinterval [1, 3], the right endpoint is 3.
In the second subinterval [3, 5], the right endpoint is 5.
In the third subinterval [5, 7], the right endpoint is 7.
So, the sample points for the three subintervals are 3, 5, and 7.
Now, we need to evaluate f(x) at each of these sample points.
For f(x) = x^2 + 1/x√ + x + 5, we have:
f(3) = 3^2 + 1/3√ + 3 + 5 = 14
f(5) = 5^2 + 1/5√ + 5 + 5 = 31
f(7) = 7^2 + 1/7√ + 7 + 5 = 59
To calculate the right Riemann sum approximation (R3), we take the sum of the areas of the rectangles formed by multiplying the function values by the width of the subinterval.
The width of each subinterval is 2.
For the first subinterval [1, 3], the area of the rectangle is f(3) * 2 = 14 * 2 = 28.
For the second subinterval [3, 5], the area of the rectangle is f(5) * 2 = 31 * 2 = 62.
For the third subinterval [5, 7], the area of the rectangle is f(7) * 2 = 59 * 2 = 118.
Finally, we sum up these areas to find the right Riemann sum approximation:
R3 = 28 + 62 + 118 = 208.
Therefore, the statement “R3 = 208” is true.
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