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(a) Using (9), estimate the area bounded by $f(x)=\frac{1}{1+x^{3}}$ and the $x$ -axis between $x=1 \text { and } x=2 . \text { [Hint: See Example } 6]$ (b) Using Riemann sums with the midpoint of each sub-interval, and $n=100$ rectangles, approximate this area using a spreadsheet. (c) Compare these results to the value of the integral $\int_{1}^{2} \frac{1}{1+x^{3}} d x$ given by your calculator.

(a) $3 / 16 < A < 3 / 8$(b) 0.24535037(c) $\frac{\ln (3 / 4)}{6}+\frac{\pi \sqrt{3}}{18} \approx 0.254353$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Campbell University

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.

04:54

Approximate the area under…

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(a) For each function defi…

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Use finite approximations …

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Write and evaluate a sum t…

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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