Integration by Riemann sums Consider the inte...

Question

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

Accepted Answer

Hello. In this video, we are going to compute the right rhyme in sum for the given definite integral with N is equal to 3 subintervals. So the definite integral that is given to us is the integral from 0 to 3 of 2 X2 + 1 DX, and we want to work with a total of 3 sub-intervals. Now, in order to approach this problem, we are going to first use the definition of the definite integral. That definition is given as the following. If we have a definite integral from A to B of F of XDX, then this is equivalent to the summation from 1 ton of F of Xi multiplied by our change in X, which is delta X. Here, delta X is equal to B minus A divided by N, and Xi is equivalent to A plus the index I multiplied by our change in XDeltaX. This is going to be the definition of the finite integral. Now, in order to approach this problem, the first thing we want to do is you want to identify our delta X, which is going to be the area of the sub-intervals. Now, delta X is equivalent to our upper boundary minus our lower boundary, divided by the amount of sub-intervals we're working with. In this case here, Delta X is going to equal to 3 minus 0 divided by 3. And this is going to simplify to just be one, so delta X is equal to 1. Now, the function given to us inside of the definite integral is F of X is equal to 2 X2 + 1. And so if we were to write this in a summation form, this is going to be the summation from Is equal to 1 to our stopping point N, which is given to us as 3 of the function 2 X2 + 1 multiplied by delta X. Now, if we were to expand this, this is going to be Delta X multiplied. By F of X1 plus F of X2, plus F of X3, and this is where X 1, X2, and X 3 are the endpoints of the boundary we're working over. Now, we need to define those right endpoints. If we were to draw a number line of the finite integral from 0 to 3, and split this up into the amount of sub-intervals we're working with, which is 3 sub-intervals, then these endpoints are going to be 012, and 3. But since we are working with right endpoints, we are going to choose the right most endpoints of this number line, which is going to be 12, and 3. And so with that being said, we are going to have to plug in these boundaries into our summation. That is going to change our summation to be Delta X is equal to F of 1 plus F of 2, plus F of 3. And so in order to simplify this, we are going to plug in our value of delta X and plug in the values of 123 into our function. Doing this is going to give us one multiplied. By 3 + 9 + 19, which is going to further simplify to be 31. So 31 is going to be the value of the summation, which is also going to be the value of the given definite integral. And with that being said, the overall solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

Key Concepts

Riemann Sums

Definite Integral

Partitioning the Interval

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