Which of the following is a right Riemann sum for ∫831+x−−−−−√ⅆx ?
To determine the right Riemann sum for the function ∫831+x−√ⅆx, we must first understand the concept of a Riemann sum and how it is calculated
To determine the right Riemann sum for the function ∫831+x−√ⅆx, we must first understand the concept of a Riemann sum and how it is calculated.
A Riemann sum is a method used to approximate the definite integral of a function over an interval. It involves dividing the interval into smaller subintervals, then evaluating the function at certain points within each subinterval. The sum of the areas of the resulting rectangles provides an approximation of the definite integral.
In this case, the integral function is ∫831+x−√ⅆx, which represents the area under the curve of the function 831+x−√.
Now, let’s explore the different types of Riemann sums: left, right, and midpoint.
1. Left Riemann Sum:
In a left Riemann sum, the height of each rectangle is determined by evaluating the function at the left endpoint of each subinterval.
2. Right Riemann Sum:
In a right Riemann sum, the height of each rectangle is determined by evaluating the function at the right endpoint of each subinterval.
3. Midpoint Riemann Sum:
In a midpoint Riemann sum, the height of each rectangle is determined by evaluating the function at the midpoint of each subinterval.
For the given function ∫831+x−√ⅆx, we are specifically looking for the right Riemann sum. This means that we will evaluate the function at the right endpoint of each subinterval.
To determine the right Riemann sum, we need to know the interval over which we are integrating. If the interval is not provided, we will assume it to be [a, b].
Now, considering the provided function, ∫831+x−√ⅆx, we can see that the interval is not given. Therefore, we cannot determine the specific right Riemann sum without additional information.
However, if an interval is given (e.g., [a, b]), we would divide the interval into smaller subintervals of equal width and evaluate the function at the right endpoint of each subinterval. Then, we would calculate the sum of the areas of the rectangles to obtain the right Riemann sum approximation.
Keep in mind that as the number of subintervals increases, the Riemann sum becomes more accurate in approximating the definite integral of a function.
Thus, without specifying the interval, it is not possible to provide a particular right Riemann sum for the function ∫831+x−√ⅆx.
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