Each of the following parts relates to finding the area underneath the curve , from x = 1 to x = 3.

D. Set up, but do not solve, the Riemann Sum which expresses the exact area underneath the curve.

CORRECTION:

Each of the following parts relates to finding the area underneath the curve , y = x^3 from x = 1 to x = 3.

D. Set up, but do not solve, the Riemann Sum which expresses the exact area underneath the curve.

Thanks.

The interval has width 2, so if there are n rectangles in the sum,

the width of each is 2/n
using right endpoints, xk = 1+(2/n)*k

          n
lim     ∑(1+(2/n)*k)^3 (2/n)
n->∞   k=1

To set up the Riemann Sum for finding the exact area underneath the curve from x = 1 to x = 3, we need to divide the interval [1, 3] into smaller subintervals and approximate the area using rectangles.

The Riemann Sum can be expressed as follows:
R = Σ f(xi)Δx

Where:
- R is the Riemann Sum
- f(xi) is the function evaluated at each x-value within the subintervals
- Δx is the width of each subinterval

In this case, we need to determine the number of subintervals and the width of each subinterval.

Let's say we want to use n subintervals. The width of each subinterval, Δx, can be calculated as:

Δx = (b - a) / n

Where:
- a is the lower limit of the interval (1 in this case)
- b is the upper limit of the interval (3 in this case)

Once we have determined the width of the subintervals, we can choose representative x-values for each subinterval. These x-values are typically chosen as the left endpoints, right endpoints, or midpoints of the subintervals.

Let's say we choose the left endpoints of each subinterval. So, for each subinterval, the representative x-value, xi, can be calculated as:

xi = a + i*Δx

Where:
- i is the index of the subinterval (starting from 0)

Finally, we can substitute the calculated values into the Riemann Sum formula to express the exact area underneath the curve. However, since we are asked to set up the Riemann Sum without solving it, we stop at this point and leave the formula as:

R = Σ f(xi)Δx

Now, you can solve the Riemann Sum by choosing appropriate values for n and evaluating the function at the representative x-values within each subinterval.