Evaluate
(lim)(sin(Pi/(n))+sin((2*Pi)/(n))+sin((3*Pi)/(n))+***+sin((n*Pi)/(n)))/(n)
by interpreting it as the limit of Riemann sums for a continuous function f defined on [0,1].
keywords: integration, integrates, integrals, integrating, double, triple, multiple

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Prove that if f is integrable on [0, 1], then
lim n !1 Z
1
0
x n f(x)dx = 0
Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition,

1. Suppose f: [a,b] is a function such that f(x)=0 for every x (a,b].
a) Let > 0. Choose n such that a + 1/n < b and |f(a)|/n <.
Let P ={a, a+1/n, b} ([a,b]). Compute (f,P) - (f,P) and show that is less than .
b) Prove

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1.
? Calculate the Taylor Polynomial and the Taylor residual for the function .
? Prove that as , for all .
? Find the Taylor series of f.
? What is the radius of convergence for the Taylor series? Justify your answer.
2.
? Let f:[0,1] be a bo

Using the Fundamental Theorem of Calculus I need to find the solution of the following problems. Can you explain how?
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Let B(x) represent the area bounded by the graph and the horizontal axis and vertical
lines at t=0 and t=x for the graph below. Evaluate B(x) for x = 1, 2, 3, 4, and 5.
f(x)=x^2,g(x)=3x,and h(x)=2/x. Evaluate each sum.
∑_(i=0)^3▒〖f(1+i)〗
sketch the function and find the smallest possible value and the l

This question has me going around in circles. I can't make the Sigma symbol on the computer, so I used the word "Sigma" instead. For (c), n is above the Sigma symbol and i=1 is below it.
(a)Find an approximation to the integral as 0 goes to 4 of (x^2-3x)dx using a Riemann sum with right endpoints and n=8.
(b)Draw a diagram

Let f be the following function with domain C = [0, 1] X [0, 1] (in two-dimensional Cartesian space):
f(x, y) = 0 on the line segments x = 0, y = 0, and x = y
f(x, y) = -1/(x^2) if 0 < y < x <= 1
f(x, y) = 1/(y^2) if 0 < x < y <= 1
Compute each iterated Riemann integral of f on C (by integrating first over x and then

This solution shows how to solve for various calculus problems, including differentiation of functions using the product rule, the quotient rule, and the chain rule, as well as how to calculate integrals.