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
Snowcat [4.5K]
3 years ago
12

Calculus question. Please complete and explain the following.

Mathematics
1 answer:
Daniel [21]3 years ago
8 0

We're approximating the area under the graph of the function f(x)=x^3+8 over the interval [-2, 2] by

  1. partitioning the integration interval into n=4 subintervals,
  2. building rectangles whose lengths are equal to the length of the corresponding subinterval and whose heights are equal to the value of f(r_i), where r_i denotes the right endpoint of the i-th subinterval, and
  3. computing the areas of each rectangle and adding these areas together.

Splitting [-2, 2] into 4 intervals gives

[-2, -1], [-1, 0], [0, 1], [1, 2]

Each subinterval has length 1. The right endpoints of the i-th subinterval, where 1\le i\le4, are given by the (arithmetic) sequence

r_i=-1+1(i-1)=i-2

The area of the rectangle over the i-th subinterval is

A_i=f(i-2)=(i-2)^3+8=i^3-6i^2+6i

and so the definite integral is approximately

\displaystyle\int_{-2}^2(x^3+8)\,\mathrm dx\approx\sum_{i=1}^4(i^3-6i^2+6i)

There are well-known formulas for computing the sums of powers of consecutive (positive) integers. The ones we care about are

\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}2

\displaystyle\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}6

\displaystyle\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}4

So we get

\displaystyle\int_{-2}^2(x^3+8)\,\mathrm dx\approx40

You might be interested in
What is 1 and 7/10 <br> as a decimal?<br> 0.17<br> 1.7<br> th<br> 0.7<br> 7.1<br> 1.07
earnstyle [38]
1.7 is the answer to the question
3 0
2 years ago
I WILL AWARD BRAINLIEST!! PLEASE HELP!! Given: ΔABC, AB = AC X ∈ AC , Y ∈AB AX = AY Prove: BX = CY m∠ABC = m∠ACB △ABX≅△____, by
EleoNora [17]

Answer:

△ABX ≅ △ACY, by reason that they are equal and congruent to each other.

I hope this answers your question.

6 0
3 years ago
2.1 ÷ 1.488 round to nearest tenth
Lorico [155]
I believe it should be 1.4 since the answer you get is 1.4112903226... you would round but on the tenths place 1.4
4 0
3 years ago
Read 2 more answers
Phillip saves $8each month. How many months will it take him to save at least $60.
Studentka2010 [4]
<span>$8 each month
</span><span>How many months will it take him to save at least $60?

60 </span>÷ 8 = 7.5

It will take Phillip 7.5 months to save up at least $60!
5 0
3 years ago
CAN SOMEONE PLEASE HELP ME???
AleksandrR [38]

Answer: The answer is 5.6d+ 2.1

Step-by-step explanation:

Add then subtract then add then multiply then subtract then divide.

5 0
2 years ago
Read 2 more answers
Other questions:
  • How to solve and answer please
    7·1 answer
  • How to find the measure of yz
    13·1 answer
  • 9e^6x-2+7=45<br>hello me, please show your work. thanks!
    11·1 answer
  • Solve: 2x - 3 = 2x - 5​
    6·1 answer
  • What is the surface area of the right prism below?
    9·1 answer
  • The domaln of the following relation R ((6, -2), (1, 2), (-3, -4), (-3, 2)} Is (1 polnt)
    5·1 answer
  • Felicia opened a bank account and made an openig deposit of $900.
    9·1 answer
  • A shipment of 15 of tennis balls weighs 1 3/4
    14·1 answer
  • The prism below is made of cubes which measure of a centimeter on one side. What is the volume?
    9·1 answer
  • A farm has 5 brown cows and 10 white cows. A fence is open and a cow escapes. What is the probability that it will be a brown co
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!