Express the limit lim n → ∞ n ∑ i = 1 ( 5 cos ( 2 π x ∗ i ) + 8 ) Δ x i over [ 1 , 4 ] as an integral.

Question

Express the limit lim n

n→∞ ∑ (5cos(2πx∗i)+8)Δxi over [1,4] as an integral.

i=1

Provide a, b and f(x) in the expression 

b

∫ f(x)dx

a

I got a= 1 and b=4, how do i solve for the function?

Answer 1

Question: Express the limit lim n→∞ ∑( (5cos(2πx∗i)+8)Δxi) (from i=1 to ∞) over [1,4] as an integral.

Answer: You're right in saying that the definite integral will be defined from 1 to 4.

Remember that each Δx_i refers to a subdivision of the interval over which you are adding rectangle areas. Each of these delta terms represent the width of these rectangles, which will become infinitesimally thin when we take the limit of the function as n approaches infinity (an infinite number of subdivisions, or rectangles). Each x_i^* term represents the value of the function achieved by each rectangle, so each f(x_i^*) represents the heights of these rectangles. When we take the limit of the function as n approached infinity, we'll be able to plug in any value in the domain of the function.

(Also, although your lower bound of integration is already 1, think separately about the definition of the definite integral, which adds up an infinite number of rectangles, starting from the first rectangle, or n=1. The n is the count of rectangles, and although it is 1 here, it's not to be confused with your integration bounds in this specific example.)

With these insights, let's rewrite the function as a definite integral:

4

∫ (5cos(2πx)+8) dx

1

In case the upload of the final answer here scatters the placement of the bounds of integration, here is another way to express the answer: "The definite integral of 5cos(2πx) + 8 with respect to x, evaluated from 1 to 4."