use reimann sums and a limit to compute the exact area under the curve of y=2x^2+1 on the interval[1,3]... i am having trouble getting the right answer which is suppose to be 58/3. I would just like to see the set up of the sum so check if am doing it right. Thank you.

The Riemann sum is the basis of definite integration.

What we are trying to do is to divide the area to be integrated into n vertical pieces of equal width (along x). Call this width h.

We then express the area of each piece according to the definition of the function to be summed. The expression is then simplified and summed from 1 to n (using the right Riemann sum).

Finally, we take the limit for n-> ∞ which should be the exact area as required.

For
f(x) = 2x²+x
to be summed from 1 to 3.
Subdivide the area into n pieces, each of width h=(3-1)/n=2/n.

The x-value for the ith piece is then
1+ih = 1+2i/n.

The corresponding y-value is:
fi=f(1+ih)=f(1+2i/n)=2(1+2i/n)²+1+2i/n.

The width of the ith piece is h=2/n.

Thus the area of the ith piece is
h*fi = (2/n)[2(1+2i/n)²+(1+2i/n)]
Expand and simplify
h*fi=(6/n+20i/n²+8i²/n³)

The total area is
A=

n
Σh*fi
i=1

n
=Σ(6/n+20i/n²+8i²/n³)
i=1

= 6 + 20n(n+1)/(2n²)+8n(n+1)(2n+1)/(6n³)

As n->∞
A=6+10+16/6=18 1/3 = 58/3

Note:
Standard summations used:
n
Σ i = n(n+1)/2
i=1

n
Σ i² = n(n+1)(2n+1)/(6)
i=1

To compute the exact area under the curve of the function y = 2x^2 + 1 on the interval [1, 3] using Riemann sums, we can break down the interval into smaller subintervals and approximate the area of each subinterval using rectangles.

Here's how you can set up the Riemann sum:

1. First, determine the width of each subinterval. We'll divide the interval [1, 3] into n equal subintervals, so the width, Δx, of each subinterval is given by Δx = (b - a) / n, where a = 1 is the left endpoint of the interval, and b = 3 is the right endpoint.

2. Next, choose a representative point within each subinterval. Let's use the right endpoint of each subinterval as the representative point for this setup. So the right endpoint of the kth subinterval is given by x_k = a + kΔx.

3. Now, calculate the height of each rectangle. For this, evaluate the function at each representative point from step 2. So the height of the kth rectangle is given by f(x_k), where f(x) = 2x^2 + 1.

4. Determine the area of each rectangle. Multiply the width (Δx) by the height (f(x_k)) for each subinterval to find the area of that rectangle.

5. Finally, sum up the areas of all the rectangles. Add up all the areas calculated in step 4 to get the approximated area under the curve.

Now, to get a more accurate approximation, we can take the limit as n approaches infinity (Δx approaches 0) to obtain the exact area. Mathematically, this is represented as:

A = lim(n→∞) Σ[k=1 to n] (f(x_k) Δx)

Now let's calculate it for the specific case of y = 2x^2 + 1 on the interval [1, 3]. Choose a large value for n, such as n = 100, to get a good approximation.

Here's the step-by-step calculation:

1. Δx = (3 - 1) / 100 = 0.02

2. For each k from 1 to 100:
- x_k = 1 + k * Δx
- Calculate f(x_k) = 2(x_k)^2 + 1

3. For each k from 1 to 100:
- Calculate the area of each rectangle:
Area_k = f(x_k) * Δx

4. Sum up the areas of all rectangles:
A = Σ[k=1 to 100] Area_k

Now, when you perform these calculations, you should get the answer as 58/3, which is approximately 19.33. If you're having trouble getting this result, double-check your calculations for each step.