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Evaluate the given definite integral.$\int_{8}^{27} \sqrt[3]{w} d w$

$$195 / 4$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Campbell University

Oregon State University

Baylor University

Idaho State 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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Okay, so we're doing the integral from 8 to 27 there's a reason why they chose those numbers of the cube root of W. D. W. And if you are good with rational exponents, that makes perfect sense that this is W to the one third power. So now we can do the anti derivative of that by adding 1 to 1 third. Remember, three thirds is equal to one, and I want to have the same denominators. So that's why it's four thirds and then multiply by the reciprocal of that new exponent from 8 to 27 now, as I plug in my bounds. First of all, it might be worthwhile to factor out the three force. Remember that that, uh, four thirds power means the cube root of 27 men to the fourth power, minus the cube root of eight, then to the fourth power. So then, from there I could think in my head that the cube root of 27 is three to the fourth. Power would be 81 cube root of eight will be two men to the fourth power be 16 well, 81 minus six would be 75 7 Must be 65 if I'm multiplying a whole number by a fraction. Just goes into the numerator. Um And so I get 65 times three is 195. The force is still there. Mhm.

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