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Use the formulas provided in this section to compute the given sum.$$\sum_{i=1}^{30}\left(4 i^{3}-2 i^{2}+11 i-12\right)$$

$$850,745$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

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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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Use Special Sum Formulas $…

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Use the formulas in exerci…

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Calculate $\sum_{j=1}^{30}…

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Use summation rules to com…

Hello. So here we have the some we have I going from 1 to 10 of I minus one times four I plus three. So we can do we can split this up as well. This some um going from I to 100 of while we did if we to foil this out this or distribute we get a four I Squared -I -3. So this is going to be equal to again to some where we have I going from 1 to 10 of Well this then becomes a four, I squared minus I minus three. Which we can then break up as the individual terms. Here is we have the sum of while four I square the four can come out front of the some nation that's going to be equal to four times the sum I going from 1 to 10 of I squared. And then we have minus two some where we have I going from 1 to 10 of just I. And then we have minus to some uh where we have I Going from 1 to 10 of just three. So therefore well this first um I square we know the formula for um the some I from one to end of I of I squared. So that's going to be we have the four out front and then that's going to give us um for Times 10 times 10 plus one times uh two times 10 plus one. And then all divided by six is our formula for the some of the first and squares and then we have minus or the sum I close 1 to 10 of just I that's going to be well 10 times 10 plus 1/2. Yeah. And then um minus well um the some I was 1 to 10 of just three. That's just three plus three plus three plus three. There's no eyes in the last some here, that's going to be equal to about 10 times so minus three times 10. Yeah. Okay. And then it's a certification here. Um we certify this, this becomes uh for this whole part here just becomes um Uh 385 so we have four times 385 and then we have minus while this here 10 times 11/2, that's 55. So we have four times 385 minus 55 then minus three times 10, so minus 30 and that should give us 1455. So the sum here is 1455. Mhm. Mhm.

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