1,450 divided by "1" /"25" 1,450 divided by "1" /"50"

If f(x, y) = esin(x + y) and D = [−π, π] × [−π, π], show the following. 1 e ≤ 1 4π2 D f(x, y) dA ≤ e The area of the region D is

Digiron [165]

I'm guessing the purpose of this exercise is to find the average value of $f(x,y)=e^{\sin(x+y)}$ over the region $D$ ,

$D=\left\{(x,y)\mid (x,y)\in[-\pi,\pi]^2\right\}$

which is

$\displaystyle\frac{\displaystyle\iint_Df(x,y)\,\mathrm dx\,\mathrm dy}{\displaystyle\iint_D\mathrm dx\,\mathrm dy}$

The denominator is the measure (area) of $D$ , which is easy to compute:

$\displaystyle\iint_D\mathrm dx\,\mathrm dy=\int_{-\pi}^\pi\int_{-\pi}^\pi\mathrm dx\,\mathrm dy=(2\pi)^2=4\pi^2$

Not to be confused with the integral in the numerator:

$\displaystyle\iint_Df(x,y)\,\mathrm dx\,\mathrm dy=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{\sin(x+y)}\,\mathrm dx\,\mathrm dy$

but this integral is difficult to compute, hence the inequalities. We have

$-1\le\sin(x+y)\le1$

$\implies\dfrac1e\le e^{\sin(x+y)}\le e$

$\implies\displaystyle\iint_D\frac{\mathrm dx\,\mathrm dy}e\le\iint_De^{\sin(x+y)}\,\mathrm dx\,\mathrm dy\le\iint_De\,\mathrm dx\,\mathrm dy$

$\implies\displaystyle\frac{4\pi^2}e\le\iint_De^{\sin(x+y)}\,\mathrm dx\,\mathrm dy\le4\pi^2e$

3 0

3 years ago

HELP!!

kupik [55]

Answer:

28 centimeters

Step-by-step explanation:

7 0

3 years ago

Marge home is Is heated by hot water circulating in pipes recently the system developed a Smalley Inn loses 1 1/5 mm every 30 mi

maksim [4K]

Answer:

why are we stilll here

Step-by-step explanation:

7 0

3 years ago

Subtract. Then, in two or more complete sentences, explained how you found the difference. In your explanation, make sure to inc

elena-s [515]

Answer:

$-1\frac{5}{2}-\frac{7}{12}=\frac{-49}{12}$

Step-by-step explanation:

Given : Expression $-1\frac{5}{2}-\frac{7}{12}$

To find : Subtract the expression ?

Solution :

Step 1 - Write the expression,

$-1\frac{5}{2}-\frac{7}{12}$

Step 2 - Write mixed form into proper form,

$-1\frac{5}{2}-\frac{7}{12}=-\frac{7}{2}-\frac{7}{12}$

Step 3 - Take least common denominator,

$-1\frac{5}{2}-\frac{7}{12}=\frac{-42-7}{12}$

Step 4 - Solve,

$-1\frac{5}{2}-\frac{7}{12}=\frac{-49}{12}$

4 0

4 years ago

1.What motivates individuals to follow the crowd

Soloha48 [4]

1. Individuals are motivated to follow the crowd because they think that that what the crowd does is normal and by doing so they will be accepted by the others.
2. Others can help us understand our behavior,by their reactions after our actions, If they are happy and satisfied this means that our behavior is "good" , and if they are angry or sad, this means our behavior is "bad".
3. It is good to follow the others if they are making good things and when they are a good example.

4 0

3 years ago