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marin [14]
3 years ago
15

Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit. lim n→∞ n i = 1 R (b) The f

ollowing formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 + 23 + 33 + + n3 = n(n + 1) 2 2
Mathematics
1 answer:
Harlamova29_29 [7]3 years ago
8 0

The summand (R?) is missing, but we can always come up with another one.

Divide the interval [0, 1] into n subintervals of equal length \dfrac{1-0}n=\dfrac1n:

[0,1]=\left[0,\dfrac1n\right]\cup\left[\dfrac1n,\dfrac2n\right]\cup\cdots\cup\left[1-\dfrac1n,1\right]

Let's consider a left-endpoint sum, so that we take values of f(\ell_i)={\ell_i}^3 where \ell_i is given by the sequence

\ell_i=\dfrac{i-1}n

with 1\le i\le n. Then the definite integral is equal to the Riemann sum

\displaystyle\int_0^1x^3\,\mathrm dx=\lim_{n\to\infty}\sum_{i=1}^n\left(\frac{i-1}n\right)^3\frac{1-0}n

=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=1}^n(i-1)^3

=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=0}^{n-1}i^3

=\displaystyle\lim_{n\to\infty}\frac{n^2(n-1)^2}{4n^4}=\boxed{\frac14}

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Expressions are mathematical statements without the equation sign

The possible integer values of q, r, and z are 3, -4 and -1

<h3>How to determine the values of the integers</h3>

The expressions are given as:

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Both expressions are equal.

So, we have:

x^2 + zx - 12 = (x + q)(x + r)

Expand

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Recall that:

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Read more about expressions at:

brainly.com/question/4344214

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Answer:

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Step-by-step explanation:

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The length of the arc is \frac{5\pi}{3} centimeters.

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