Question

Evaluate the definite integrals. $$\int_{0}^{2} x^{3} d x$$

   Evaluate the definite integrals. 
$$\int_{0}^{2} x^{3} d x$$

Calculus: A Complete Course

Calculus: A Complete Course

Robert Adams ,… 9th Edition

Chapter 5, Problem 1

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Solved by Expert Tyler Moulton

10/06/2021

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The antiderivative of $x^{3}$ is $\frac{x^{4}}{4}$. Show more…

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Evaluate the definite integrals. $$\int_{0}^{2} x^{3} d x$$

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Key Concepts

Integration Power Rule

The integration power rule is a method for finding the antiderivative of functions of the form x^n, where n is a real number not equal to -1. It states that the integral of x^n with respect to x is x^(n+1)/(n+1), plus a constant of integration in the case of indefinite integrals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes the connection between differentiation and integration. It states that if a function is continuous over an interval, then its definite integral over that interval can be computed by finding an antiderivative and evaluating it at the endpoints, subtracting the value at the lower limit from that at the upper limit.

Definite Integral

A definite integral computes the accumulation of a quantity, often interpreted as the area under a curve, between two specified limits. It yields a numerical value that represents the total accumulation over the interval, rather than a general antiderivative function.

Antiderivative

An antiderivative of a function is another function whose derivative is equal to the original function. Finding an antiderivative is essential in integration because it allows for the evaluation of definite integrals using the differences between the values at the upper and lower limits.

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