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
Tom [10]
2 years ago
15

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves

about the x-axis. (tried this question a lot and I am just missing something)(plz help)
x = −3y2 + 9y − 6, x = 0
Mathematics
1 answer:
Vadim26 [7]2 years ago
6 0

The expression on the left side describes a parabola. Factorize it to determine where it crosses the y-axis (i.e. the line x = 0) :

-3y² + 9y - 6 = -3 (y² - 3y + 2)

… = -3 (y - 1) (y - 2) = 0

⇒   y = 1   or   y = 2

Also, complete the square to determine the vertex of the parabola:

-3y² + 9y - 6 = -3 (y² - 3y) - 6

… = -3 (y² - 3y + 9/4 - 9/4) - 6

… = -3 (y² - 2•3/2 y + (3/2)²) + 27/4 - 6

… = -3 (y - 3/2)² + 3/4

⇒   vertex at (x, y) = (3/4, 3/2)

I've attached a sketch of the curve along with one of the shells that make up the solid. For some value of x in the interval 0 ≤ x ≤ 3/4, each cylindrical shell has

radius = x

height = y⁺ - y⁻

where y⁺ refers to the half of the parabola above the line y = 3/2, and y⁻ is the lower half. These halves are functions of x that we obtain from its equation by solving for y :

x = -3y² + 9y - 6

x = -3 (y - 3/2)² + 3/4

x - 3/4 = -3 (y - 3/2)²

-x/3 + 1/4 = (y -  3/2)²

± √(1/4 - x/3) = y - 3/2

y = 3/2 ± √(1/4 - x/3)

y⁺ and y⁻ are the solutions with the positive and negative square roots, respectively, so each shell has height

(3/2 + √(1/4 - x/3)) - (3/2 - √(1/4 - x/3)) = 2 √(1/4 - x/3)

Now set up the integral and compute the volume.

\displaystyle 2\pi \int_{x=0}^{x=3/4} 2x \sqrt{\frac14 - \frac x3} \, dx

Substitute u = 1/4 - x/3, so x = 3/4 - 3u and dx = -3 du.

\displaystyle 2\pi \int_{u=1/4-0/3}^{u=1/4-(3/4)/3} 2\left(\frac34 - 3u\right) \sqrt{u} \left(-3 \, du\right)

\displaystyle -12\pi \int_{u=1/4}^{u=0} \left(\frac34 - 3u\right) \sqrt{u} \, du

\displaystyle 12\pi \int_{u=0}^{u=1/4} \left(\frac34 u^{1/2} - 3u^{3/2}\right)  \, du

\displaystyle 12\pi \left(\frac34\cdot\frac23 u^{3/2} - 3\cdot\frac25u^{5/2}\right)  \bigg|_{u=0}^{u=1/4}

\displaystyle 12\pi \left(\frac12 u^{3/2} - \frac65u^{5/2}\right)  \bigg|_{u=0}^{u=1/4}

\displaystyle 12\pi \left(\frac12 \left(\frac14\right)^{3/2} - \frac65\left(\frac14\right)^{5/2}\right) - 12\pi (0 - 0)

\displaystyle 12\pi \left(\frac1{16} - \frac3{80}\right) = \frac{12\pi}{40} = \boxed{\frac{3\pi}{10}}

You might be interested in
(Due soon, please hurry!)
Ludmilka [50]

Answer:

I Dont Know

Step-by-step explanation:

8 0
2 years ago
Read 2 more answers
In a first -aid kit the ratio of large bandages is 3 to 2. Based on this ratio, how many large bandages are in the kit if there
11Alexandr11 [23.1K]

Answer:

  48

Step-by-step explanation:

The total number of ratio units is 3+2 = 5, and the 3 ratio units representing large bandages make up 3/5 of that total. Thus, large bandages will make up 3/5 of the total number of bandages:

  3/5×80 = 48 . . . . number of large bandages in the kit

7 0
3 years ago
You may have noticed that in practice problems related to order, logarithms are usually just "log". As you know from algebra, th
Usimov [2.4K]

Explanation:

A logarithm in one base is a constant multiple of a logarithm in any other base. Any "order of ..." specification does not include the applicable constant multiplier or the smaller order terms that may be required for an exact computation.

The concept of "order of" is similar to the concept of the degree of a polynomial. Knowing the degree of a polynomial tells you something about the "end behavior" as the function argument gets large. The specifics of the scale factor and lower-degree terms become largely irrelevant.

4 0
3 years ago
A store has apples on sale for $5.00 for 2 pounds. If an apple is approximately 5 ounces, how many apples can you buy for $30.00
jeyben [28]
R $30.00. Since 1 pound = 16 ounces, so there are about apples in each pound. You can buy approximately apples for $3
<span>r $30.00? Complete the explanation. The cost per pound of the app</span>
8 0
3 years ago
HELP?!?
spin [16.1K]

Step-by-step explanation:

hear is answer in attachment

6 0
3 years ago
Read 2 more answers
Other questions:
  • Suppose that the cost of renting a snowmobile is 37.50 for 5 hours.
    9·2 answers
  • What is the answer to 8(g+2)=2(23-g)
    11·2 answers
  • five friends split the cost of parking at an amusement park. Each of them also buys a $30 ticket.write an algebraic expression t
    15·1 answer
  • Can you please help me whit this graph to put the point we’re it belongs
    9·1 answer
  • Y and x have a proportional relationship and y =5 when x = 4 what is the value of x when y =8
    13·2 answers
  • Which expression is represented by the phrase "the square of y decreased by the quotient of 28 and 7?
    12·2 answers
  • Which of the following is an undefined term? (6 points)
    11·1 answer
  • Rodger put $1,000 in a bank account that pays 5% annual simple interest. At the end of four years, how much interest has he earn
    8·1 answer
  • There is 1/3 gallon of water in a 3-gallon container. What fraction of the container is filled? Draw a tape diagram to represent
    7·1 answer
  • HELP PLEASE!!!<br><br>Answer: ?​
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!