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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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3 years ago
Weekly wages at a certain factory are normally distributed with a mean of $400 and a standard deviation of $50. Find the probabi
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$80

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2 years ago
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4 0
2 years ago
The length of a rectangle is 6 inches more than the width. The perimeter is 40 inches. Find the length and the width of the rect
eduard

Answer:

lenght = 13 inches

width = 7 inches

Step-by-step explanation:

p= 2(l+b)

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width be y

so x=y+6

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3 years ago
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Answer:

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