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(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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Calculus 1 / AB
Chapter 5
Integrals
Section 1
Areas and Distances
Integration
Harvey Mudd College
Baylor University
Idaho State 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.
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