Let f(x) =

dmitriy555 [2]

In Cartesian coordinates, the region (call it $R$ ) is the set

$R = \left\{(x,y,z) ~:~ x\ge0 \text{ and } y\ge0 \text{ and } 2 \le z \le 4-x^2-y^2\right\}$

In the plane $z=2$ , we have

$2 = 4 - x^2 - y^2 \implies x^2 + y^2 = 2 = \left(\sqrt2\right)^2$

which is a circle with radius $\sqrt2$ . Then we can better describe the solid by

$R = \left\{(x,y,z) ~:~ 0 \le x \le \sqrt2 \text{ and } 0 \le y \le \sqrt{2 - x^2} \text{ and } 2 \le z \le 4 - x^2 - y^2 \right\}$

so that the volume is

$\displaystyle \iiint_R dV = \int_0^{\sqrt2} \int_0^{\sqrt{2-x^2}} \int_2^{4-x^2-y^2} dz \, dy \, dx$

While doable, it's easier to compute the volume in cylindrical coordinates.

$\begin{cases} x = r \cos(\theta) \\ y = r\sin(\theta) \\ z = \zeta \end{cases} \implies \begin{cases}x^2 + y^2 = r^2 \\ dV = r\,dr\,d\theta\,d\zeta\end{cases}$

Then we can describe $R$ in cylindrical coordinates by

$R = \left\{(r,\theta,\zeta) ~:~ 0 \le r \le \sqrt2 \text{ and } 0 \le \theta \le\dfrac\pi2 \text{ and } 2 \le \zeta \le 4 - r^2\right\}$

so that the volume is

$\displaystyle \iiint_R dV = \int_0^{\pi/2} \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta \, dr \, d\theta \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta\,dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} r((4 - r^2) - 2) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} (2r-r^3) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \left(\left(\sqrt2\right)^2 - \frac{\left(\sqrt2\right)^4}4\right) = \boxed{\frac\pi2}$

3 0

1 year ago

Harvey the wonder hamster can run 3 1/6 km in 1/4 hour harvey runs at a constant rate what is his average speed in kilometers pe

vampirchik [111]

1/4 times 4 is 1 hour so
3 1/6 times 4 is 12 2/3

3 0

3 years ago

4. Why studying central tendency alone is insufficient and calculating dispersion is always needed in both descriptive and infer

ira [324]

Answer:

Check below for the answer and explanation.

Step-by-step explanation:

Studying the central tendency alone is not sufficient because apart from calculating the value of the central point of a group of data ( which is what the measure of central tendency does), it is important to also understand the spread of these data about the average(mean) value.The measure of dispersion will help us to know the range of error that is recorded in both descriptive and inferential statistics and this will enable the statistician to assess the validity of the data generated from the experiment performed.

A small value of standard deviation indicates that each of the values in the dataset is close to the average (mean) value.

4 0

3 years ago

Angela and Jayden were at track practice. The track is 2/5 kilometers around. Angela ran 1 lap in 2 Minutes. Jayden ran 3 laps i

I am Lyosha [343]

Answer:

Step-by-step explanation:

Angel ran 1 lap in 2 minutes.

Jayden ran 3 laps in 5 minutes.

How many minutes does it take Angel to run one kilometer? What about Jayden?

How far does Angel run in one minute? What about Jayden?

Who is running faster? Explain your reasoning.

IM Commentary

Parts (a) and (b) of the task ask students to find the unit rates that one can compute in this context. Part (b) does not specify whether the units should be laps or km, so answers can be expressed using either one.

The purpose of part (c) is to give students an opportunity to make use of the unit rates that they found in parts (a) and (b). While it is possible for students to solve part (c) in other ways, the solution shown represents the kind of reasoning with unit rates that 7th graders should be able to do. It is important to note that the answer can be determined using different unit rates as long as the reasoning behind it is correct.

Solution

We can create a table that shows how far each person runs for a certain number of laps:

Number of laps Number of km

1 25

2 45

3 65

We can see from the table that 1 km is exactly half way between 2 and 3 laps. So it will take 2.5 laps to run 1 km.

Since it takes Angel 2 minutes to run 1 lap, she will take

2.5 laps1 km⋅2 minutes1 lap=5 minutes1 km.

So it takes Angel 5 minutes to run 1 km.

Since it takes Jayden 5 minutes to runs 3 laps, she runs 1 lap in 53 minutes. Thus, it takes Jayden

2.5 laps1 km⋅5 minutes3 laps=52⋅53 minutes/km=256 minutes/km=416 minutes/km.

So it takes Jayden 416 minutes to run 1 km.

Angel runs 1 lap in 2 minutes so she runs 12 lap in 1 minute. Since 1 lap is 25 km, 12 lap is 15 km. So she also runs 15 km in one minute.

Since Jayden runs 1 lap in 53 minutes, she will run 35 laps in 1 minute. Since Jayden runs 1 km in 256 minutes, she will run 625 km in 1 minute.

Jayden runs the same distance in less time than Angel (alternatively, Jayden runs farther in the same time than Angel), so Jayden is running faster than Angel.

4 0

2 years ago

Find the perimeter of the parallelogram with these vertices. (-2.2).(-2.-2), (6.-1). (6.-5) Give an exact answer (not a decimal

nasty-shy [4]

Answer:

16units

Step-by-step explanation:

5 0

3 years ago