Integration by Riemann sums Consider the inte...

Question

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

(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .

Accepted Answer

Hello. In this video, we are asked to write a right to rhyme in some in some nation notation for the given definite integral in terms of a positive integer N. So in order to approach this problem, we are going to have to 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 some function in terms of X, then this is equivalent to a summation from 1 to some positive integer n of F of Xi multiplied by delta X. Here, delta X represents the change in the subintervals, and that is going to be defined as B minus A divided by N. Also, our variable X sub I is going to be defined as A plus the index I multiplied by Delta X. So, in order to approach this problem, we need to go ahead and identify the different pieces we need for the given summation. Now, the definite integral given to us is the integral from 0 to 2 of 5 X minus 4DX. By looking at the bounds of the definite integral, we can tell that our boundary or interval from A to B is going to be defined from 0 to 2, and we can use this to define our delta X. Delta X is going to be B minus A divided by N, which is going to be 2 minus 0 divided by N. and this is going to further simplify to be 2 divided by N. So this is going to be our value of delta X. Next, what we need to do is we need to define our Xi terms to plug into the function. Now, X sub I, again, is A plus I delta X. Here we know that A is 0, so we're going to have 0 plus I multiplied by the value of delta X, which we just calculated to be 2 divided by N. And this is going to further simplify to be 2 I divided by N. Now that we have our delta X and our X sub I, we can now put together the summation for the given definite integral. Now that is going to be the summation starting from 1 to some positive integer N and F of X sub I is going to be defined as the function 5X minus 4, with delta X plugged into it. That is going to give us 5 multiplied by 2 I divided by N minus 4, and this will be multiplied by delta X afterward, which is 2 divided by N. Multiplying 5 and 2 I divided by N together will give us 10 I divided by N. and so the final form of the summation is going to be the summation from Is equal to 1 ton of 10 I divided by N minus 4 multiplied by 2 divided by N. and this is going to be the summation that represents the given definite integral. And with that being said, the overall solution to this problem is going to be a. 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.

(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .

Key Concepts

Riemann Sums

Definite Integral

Summation Notation

Watch next