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Let $ A $ be the area under the graph of an incre…

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Problem 26 Medium Difficulty

(a) Use Definition 2 to find an expression for the area under the curve $ y = x^3 $ from 0 to 1 as a limit.

(b) The following formula for the sum of the cubes of the first $ n $ integers is proved in Appendix E. Use it to evaluate the limit in part (a).

$$ 1^3 + 2^3 + 3^3 + \cdots + n^3 = \biggl[ \frac{n(n + 1)}{2} \biggr]^2 $$


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Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 5

Integrals

Section 1

Areas and Distances

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Integration

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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.

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Area Under Curves - Overview

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.

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Watch More Solved Questions in Chapter 5

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Video Transcript

So try to figure out this inter crow. And I mean, we right out of general formula for this specific proper and are a zero piece one and the F of axis x Q. So this gives us one of run some eye from one, two and eight zero piece one. So this high over on Q so high to over on cue and by the formula given impart, be so this. So this will be answered ofthe party and by this given form last. We know that this you can compare the leave me like this We fact at once or over a cube. So we are one over on to the fourth and we some over I too, from one toe on which it's given by this formula. So is for on square on plus one square. And this limits one over for people's we have for Lamia enumeration denominator. They have the same decree degree course for so the limit is just a racial off coefficient of the highest order term in this case will be one over four

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40:35

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