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 to illustrate the approximation in part (a).
(c)Use this equation (the integral as a goes to b of f(x)dx=the limit as n goes to infinity of Sigma f(xi) delta x) to evaulate the integral as 0 goes to 4 of ((x^2)-3x)dx.
(d)Interpret the integral in part (c) as a difference of areas and illustrate it.

Solution Preview

Ok First draw this function from x = 0 to x = 4 I get f(x) values 0,-2,-2,0, and 4.

now divide the x axis into 8 equal sections for 0 to 4 The area of interest is between. The x axis and the curve. The area under the x axis is a negative area and the area above the x- axis ...

Solution Summary

Riemann sums are explained. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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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,

Please see the attached file for the fully formatted problems.
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

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

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Write a Riemannsum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly.

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 Riemannsums for a continuous function f defined on [0,1].
keywords: integration, integrates, integrals, integrating, double, triple, multiple

I can't figure out exactly how to formulate a riemannsum.
For example, when given y=x+2; [0,1], and told to "find the area of the region under the curve y=f(x) over the interval [a,b]. To do this, divide [a,b] into n equal subintervals, caluculate the area of the cooresponding circumscribed polygon, and then let n go to infin

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