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Evaluate the given definite integral.$$\int_{8}^{27} \frac{1}{\sqrt[3]{t^{2}}} d t$$

3

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Missouri State University

Oregon State University

Harvey Mudd College

Baylor University

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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We're looking at the integral from 8 to 27 of one over the cube root of T squared DT. Now I would have my students rewrite this first of all as a negative exponent, because it's in the denominator and the square goes into the numerator of a rational exponents and the cube goes in the denominator. So now the problem is easier to evaluate the anti derivative, because if you add 12 negative two thirds, maybe just do some scratch work off to the side frame where that needs it. Adding one would be the same thing as adding three thirds. So from there you can say negative two plus three is one third. But then, if you multiply by the reciprocal that new excellence of three times that and you can confirm that that's correct by taking the derivative of that, it's the same from 8 to 27. Now, as a friendly reminder, this means, But first of all, male. Just factor out the three. It just means three times the cube root of 27 because you plug in your bounce minus the cube root of H. So as I'm looking at that well, That's pretty straightforward. The cube root of 27 is three Cuba motivators to three minus two is one and one times anything is that thing. So one times three is three the correct answer.

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