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Evaluate the given definite integral.$$\int_{1}^{64}(\sqrt{w}-\sqrt[3]{w}) d w$$

$$1793 / 12$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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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Evaluate the integral.

we're evaluating the integral from 1 to 64. Yeah, and I would rewrite the square root as to the one half power and same thing with the cube root right to the one third power. So then, from here, it makes more sense to do the anti derivative where you add one to the exponents, or one half plus one is three halves and then multiply by the reciprocal of that new experiment. Same thing with this. W to the one third. If I add 1 to 1 third, that's four thirds. Please double check my work and then multiply by the reciprocal of that new exponent. And I go from 1 to 64. Uh, now they strategically picked out 64. Because as I plugged that into your balance, remember, this is the square root cubed, and this is the cube root to the fourth. So that's why I mean, they strategically pick those numbers because the square is 64 is eight and I'm gonna use a calculator because I figure what cubed is often is 512 and, uh, the cube root of 64 is 44 to the fourth. Power is 256 And then you also have to plug in one and for all these. But what's nice about plugging in one is one to any power is just one. So those coefficients are still there. And at this point, I would really just go to a calculator and type this away Just because I'm not that good at divided by three or even divided by four. Um, if I was smart, I would have cancelled out one of those fours and just send forward to the third power. But either way, you go about doing this problem. Uh, you should get the answer of 17. 93. And again, I just played into my calculator. 12. Um and you might have a teacher. That is okay with decimals. Otherwise, you know, using a calculator, whatever equivalent answer that is. You know, you might see an answer of 149.416 and the sixth repeating if you do that correctly,

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