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lapo4ka [179]
3 years ago
10

(a) 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 Correct

: Your answer is correct. (b) The following 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:
Luba_88 [7]3 years ago
6 0

Splitting up the interval of integration into n subintervals gives the partition

\left[0,\dfrac1n\right],\left[\dfrac1n,\dfrac2n\right],\ldots,\left[\dfrac{n-1}n,1\right]

Each subinterval has length \dfrac{1-0}n=\dfrac1n. The right endpoints of each subinterval follow the sequence

r_i=\dfrac in

with i=1,2,3,\ldots,n. Then the left-endpoint Riemann sum that approximates the definite integral is

\displaystyle\sum_{i=1}^n\frac{{r_i}^3}n

and taking the limit as n\to\infty gives the area exactly. We have

\displaystyle\lim_{n\to\infty}\frac1n\sum_{i=1}^n\left(\frac in\right)^3=\lim_{n\to\infty}\frac{n^2(n+1)^2}{4n^3}=\boxed{\frac14}

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Here's link to the answer:

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diamong [38]

Move the decimal to the right (you go to the right since the exponent is positive, if the exponent was negative then you would move left) seven times, giving you 9,300,000 miles (for Earth). You then do the same with the next equation, giving you 140,000,000 miles (for Mars).

Finally, to find the official number of miles, you simply work the equation 140,000,000 - 9,300,000 .. giving you 130,700,000 miles.


If your teacher wants you to put it back into exponential form, then you first make the number into a decimal that is greater than one and less than ten.


1.30700000


Then, what you want to do is count all of the zeros as a single exponent, giving you 5, but we're not done yet. To get the correct answer, you have to also count the numbers until you reach the decimal. Remember to take out the zeros, because they're not really necessary anymore!


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expeople1 [14]

Answer: 17 miles

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Step-by-step explanation:

4 0
3 years ago
What is the slope of the line with points (-2,0) and (8,0)
Natali5045456 [20]
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3 years ago
By driving 8 mph faster than Bob, John can make a 230 mile trip in one half hour less. How fast does Bob drive on the trip? Roun
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Answer: The speed of Bob is 56.8 km/hr.

Step-by-step explanation:

Let the speed of Bob be 'x'.

Let the speed of John be 'x-8'.

Distance covered = 230 miles

time = 1\dfrac{1}{2}=\dfrac{3}{2}\ hr

According to question, we get that

\dfrac{230}{x}-\dfrac{230}{x+8}=\dfrac{3}{2}\\\\230\dfrac{x+8-x}{x(x+8)}=\dfrac{3}{2}\\\\\dfrac{230\times 8}{x^2+8x}=\dfrac{3}{2}\\\\\dfrac{1840}{x^2+8x}=\dfrac{3}{2}\\\\1840\times 2=3x^2+24x\\\\3680=3x^2+24x\\\\3x^2+24x-3680=0\\\\x\approx 56.8\ km/hr

Hence, the speed of Bob is 56.8 km/hr.

6 0
2 years ago
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