Which statements must be true for a parallelogram to be a square?

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

Vadim26 [7]

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.

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

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

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

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

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

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

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

6 0

2 years ago

What is 2.5 = 75 please help

arlik [135]

Answer: 3.333333333333% as a percent 3.25 as a decimal

7 0

3 years ago

tabita bought peppers that cost 0.79 per pound she paid 3.95 for the peppers how many pounds did she buy

Akimi4 [234]

5 pounds because 3.95/0.79 =5

6 0

4 years ago

If (8 to the power of 9)=8 to the power of 18 , what is the value of p?

blondinia [14]

The answer to this question is 8 to the power of 18 or 8 to the power of 9

4 0

4 years ago

My next zoom class starts in a few so i need this 'riddle' done. tysm

charle [14.2K]

Answer:

5761

Step-by-step explanation:

First, let's find the digit in the tenths place. The quotient of 10 and 2 is 5 because 10/2 = 5. Next, the digit in the hundredths place is the product of 3.5 and 2. 3.5 * 2 = 7. The number in the thousandths place is the sum of the digit in the tenths place and the ten thousandths place, so we have to find the digit in the ten thousandths place first. It says that every number in the code is divisible by the digit in the ten thousands place. The factors of 5 are 1,5 and the factors of 7 are 1,7. The only number that is the same in both is 1, so the last digit must be 1. Knowing that information, we can add the digit in the tenths place (5) to the number in the ten thousandths place (1) to get the third digit (6). Now we place those digits in order: tens (5), hundredths (7), thousandths (6), and ten thousandths (1) to get 5761.

4 0

4 years ago