I am totally stumped... PLEASE if anyone can help we are going nuts...lol

Use the Rieman sum definition of the definite integral to show that
3¡ò(there is a little "b" at top and little "a" at bottom
3¡ò x©÷dx = b©ø-a©ø
Here,for simplicity we assume b>a>0. Specify your mesh/grid Xi, ¥Äx, and your choice of sample points x*/i
my note: i did not know how to put the asterick over the i so i showed it as */i above
and again I WILL LOVE YOU IF SOMEONE CAN HELP ME OMG HELPPPPPPPPPP lol I feel so stupid

I'm stumped by the cryptic symbols. Try the notation

integral[a,b]
x_i or xi for x sub i

Whatever the function, the Riemann sum is evaluated by approximating the actual area with a set of rectangles, whose upper corners are points on the curve.

Review your text, and maybe take a look here:

http://mathworld.wolfram.com/RiemannSum.html

To use the Riemann sum definition of the definite integral to show that 3∫∫ x^2 dx = b^3 - a^3, you need to break down the interval [a, b] into subintervals and approximate the area under the curve using rectangles.

1. Start by dividing the interval [a, b] into n equal subintervals. The width of each subinterval is given by Δx = (b - a)/n.

2. Now, for each subinterval, choose a sample point x*/i. This can be any point within the subinterval [x_i-1, x_i], where x_i-1 represents the left endpoint and x_i represents the right endpoint of each subinterval. Let's denote the sample points as x*_1, x*_2, ..., x*_n.

3. Next, construct rectangles over each subinterval by using the height of the function values at the sample points. In this case, the function is f(x) = x^2. Therefore, the height of each rectangle will be given by f(x*_i) = (x*_i)^2.

4. Calculate the area of each rectangle. This is done by multiplying the width of the subinterval Δx with the height of the function at the respective sample point. The area of the i-th rectangle is Ai = Δx * f(x*_i) = Δx * (x*_i)^2.

5. Finally, sum up the areas of all the rectangles to get the Riemann sum. The Riemann sum for this particular function is expressed as:

R(n) = Σ Ai = Σ Δx * (x*_i)^2.

6. As you increase the number of subintervals, n, the Riemann sum will get closer to the actual value of the definite integral. In the limit as n approaches infinity, the Riemann sum converges to the definite integral:

∫∫ x^2 dx = lim(n→∞) R(n).

7. To find the definite integral of x^2 over the interval [a, b], substitute the appropriate values into the formula:

3∫∫ x^2 dx = lim(n→∞) Σ Δx * (x*_i)^2.

8. Simplify the expression by evaluating the limits and computing the sum. This will give you the result:

3∫∫ x^2 dx = b^3 - a^3.

By following these steps and applying the Riemann sum definition, you can show that 3∫∫ x^2 dx = b^3 - a^3.