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Given $f(x)=\sqrt[3]{x},$ (a) find the area bounded by this curve and the $x$ -axis for $0 \leq x \leq 1 .$ Hint: Find an irregular partition that will enable you to determine this area: note, that the endpoints should be: $0,1 / n^{3}, 8 n^{3},$ and so on, why? (b) Suppose you used a regular partition, what sum do you obtain?(c) Comparing (a) and(b) show that $\lim _{n \rightarrow \infty} \frac{1}{n^{\frac{1}{3}}} \sum_{k=1}^{n} k^{\frac{1}{3}}=\frac{3}{4}$.

(a) $3 / 4$(b) $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{i}{n}\right)^{1 / 3} \frac{1}{n}$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Missouri State University

Harvey Mudd College

Baylor University

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

01:26

(a) For each function defi…

01:44

Let $P$ be a regular parti…

07:42

Find a formula for the Rie…

05:14

For the functions find a f…

mhm for this problem we are asked to for the function f of X equals root X plus one. To find on the interval from 0 to 3, use a regular partition to form a Riemann sums. Uh in the form expressed above the first thing that we want to do is figure out what our delta act should be one second here. We want to figure out what our delta X is. Well we have interval from 0 to 3, so that's just going to be three over N. So we are asked to express this as a riemann sum. So that is going to be the sum from I equals one up to n of while our function is route X plus one. So that's going to be route you I plus one Times Delta X, So Times three Over N. Put that in brackets. Then for part B were asked to express the limit as an approaches infinity of the Riemann sums as a definite integral. So I'm just going to put limit as n approaches infinity of the thing above That is going to equal the integral from zero up to three of the square root of X plus one dx. And lastly for part C were asked to use a computer algebra system to evaluate that's definite integral. So let's see here The final result there is going to be that that is equal to 14/3

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