Riemann Sum help!?

Question

Find the Riemann sum associated with f(x)=3 x^2 +1 , n=3 and the partition
X0=−1, X1=2, X2=3, X3=5, of [−1,5]
(a) when Xk* is the right end-point of [Xk−1,Xk].________?? .

(b) when Xk* is the mid-point of [Xk−1,Xk]._________?? .



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NOTE: The digits such as 0,1,2,3 directly next to the X's are subscripts where im identifying the partition. And the k directly next to the Xs are also subscripts!

Answer 1

Somewhat unusually, your divisions are unequal. In any event, all you have to do is evaluate f(x) at the appropriate points, multiply by the width of the partition, and add the results.

I will demonstrate with the first partition only. It extends from -1 to 2, so it has a width of 3.

In a), where you evaluate at the right end-point, you evaluate at x1 = 2. f(2) = 3*2^2 + 1 = 3*4 + 1 = 12 + 1 = 13. Multiplying by the width gives you 13*3 = 39.

In b), where you evaluate at the midpoint, you evaluate at (x1 - x0) / 2 = (2 - (-1)) / 2 = 3 / 2 = 1.5. f(1.5) = 3*1.5^2 + 1 = 3*2.25 + 1 = 6.75 + 1 = 7.75. Multiplying by the width gives you 7.75*3 = 23.25.

Repeat this process for the second and third partitions as well. Note that the other partitions have different widths.

Answer 2

You will be partitioning the interval [-1,5] into n=3 pieces. Δx = (b-a)/n = 2.

You are essentially adding up the areas of the 3 rectangles with heights given by the function at various x-values. For the right endpoint Reimann sums, your x inputs are (x0 + Δx*k) = -1+2k, or 1, 3 and 5.

So the right endpoint sum = 2[f(1)+f(3)+f(5)]
The sum expressed in sigma notation is:

Σ from k=1 to 3 of (f(-1+2k)*Δx
= Σ from k=1 to 3 of (3(-1+2k)²+1)*2

For the midpoint sum, the values of the three x's are 0, 2 and 4. These can be generalized by (x0 + .5Δx*k) = (-1 + .5*2*k) = (-1+k).

Hence, the summation is

Σ from k=1 to 3 of (3(-1+k)²+1)*2

= 2[f(0)+f(2)+f(4)]

The definition of the integral, by the way, is defined as the sum of these rectangles as n, the # of rectangles, approaches infinity.