1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
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}

You might be interested in
Pls help me with this no links please and Thank you
den301095 [7]
The top graph is correct because an x=# will create a vertical line at that #
4 0
3 years ago
Read 2 more answers
If 1=2 what conclusions can u draw about m 3 and m 4
BlackZzzverrR [31]

Answer: second one

Step-by-step explanation:

5 0
3 years ago
An angle bisector always creates two acute angles. find a counterexample to show that the conjecture is false.
sweet-ann [11.9K]
An acute angle is an angle that is less than 90°. An angle bisector is a ray drawn along an angle that bisects it into two equal and adjacent parts. Now, if the total angle is, say 270°, which is more than a half circle, it would result to two 135-degree angles. In this case, the angle is no longer acute, but obtuse.
5 0
3 years ago
At the fair, Kate found a strange machine with a sign on it labeled, "Enter a number." When she pushed the
Elanso [62]

Answer:

it reduce the number by 6 mark me brainlist

7 0
3 years ago
WILL MARK BRANLIEST 50 POINTS PLEASE HELP ( CORRECT ANSWER )<br><br> Simplify. 30k2 + 6k 5k + k
Vera_Pavlovna [14]
Simplified answer would be 60k^2+k

and you get that answer by combining like terms, Good Luck! 

8 0
3 years ago
Read 2 more answers
Other questions:
  • H varies directly as L. If H=20 when L=50, determine H when L=30
    9·1 answer
  • colise has 2 vases of flowers that each hold 10 red and 5 yellow roses. Michelle has the same total number of roses but in 3 vas
    7·1 answer
  • The pyramid shown has a square base that is 24 centimeters on each side. The slant height is 16 centimeters. What is the lateral
    15·2 answers
  • What is the effect on the graph of the function f(x) = |x| when f(x) is changed to f( 3/4 x)?
    15·1 answer
  • Question 1 (2 points)
    13·1 answer
  • The scale of a map is 1 inch: 250 miles. City A is 378 miles from City B. To the nearest tenth, how far is its distance on the m
    5·2 answers
  • A recipe for beef stew calls for 1 pound of beef and 3 potatoes. The recipe is doubled to include 2 pounds of beef and 6 potatoe
    5·2 answers
  • Roger Brown works for the sanitation department. He earns a salary of $721.00 biweekly. His boss gave him
    12·1 answer
  • Solve the simultaneous equations <br> 3 x + 2 y = 18 <br> 3 x + y = 12
    11·1 answer
  • A bakery sold a total of 300 cupcakes in a day, and 252 of them were chocolate flavored. what percentage of cupcakes sold that d
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!