The given iterated integral is [tex]\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx[/tex]. The value of this integral is evaluated as 2/3.
The iterated integral is obtained by repeatedly integrating multivariate functions with a single variable each time. The given iterated integral is [tex]\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx[/tex]. This integral is evaluated in the following order [tex]\int dy[/tex], [tex]\int dz[/tex], and [tex]\int dx[/tex].
First, integrate the above expression with respect to y. Then, substitute the limit values, and we get,
[tex]\begin{aligned}\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx&=\int\limits^{\pi}_b\int\limits^1_0z\sin(x)\left[\int\limits^{\sqrt{1-z^2}}_0 \;dy\right]\;dz \;dx\\&=\int\limits^{\pi}_b\int\limits^1_0z\sin(x)[y]^{\sqrt{1-z^2}}_{0}\;dz \;dx\\&=\int\limits^{\pi}_b\int\limits^1_0z\sin(x)\left(\sqrt{1-z^2}\right)\;dz \;dx\end{aligned}[/tex]
Now, integrating with respect to z, we get,
[tex]\begin{aligned}\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx&=\int\limits^{\pi}_b\int\limits^1_0\sin(x)\;z\left(\sqrt{1-z^2}\right)\;dz \;dx\\&=-\frac{1}{2}\int\limits^{\pi}_b\sin x\left[\int\limits^1_0-2z\left(\sqrt{1-z^2}\right)\;dz\right]\;dx\end{aligned}[/tex]
Substitute, 1-z²=u and -2zdz=du in the above expression, and we get,
[tex]\begin{aligned}\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx&=-\frac{1}{2}\int\limits^{\pi}_b\sin x\left[\int\limits^0_1\left(\sqrt{u^2}\right)\;du\right]\;dx\\&=-\frac{1}{2}\int\limits^{\pi}_b\sin x\left[\frac {u^{3/2}}{3/2}\right]^0_1\;dx\\&=\frac{1}{3}\int\limits^{\pi}_b\sin x\left[u^{3/2}\right]^1_0\;dx\\&=\frac{1}{3}\int\limits^{\pi}_b\sin x\;dx\end{aligned}[/tex]
Finally, integrating with respect to x and we get,
[tex]\begin{aligned}\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx&=\frac{1}{3}\int\limits^{\pi}_b\sin x\;dx\\&=\frac{1}{3}[-(-1)-(-(1))]\\&=\frac{1}{3}[2]\\&=\frac{2}{3}\end{aligned}[/tex]
The required answer for the given integral is 2/3.
The complete question is -
Evaluate the iterated integral [tex]\int\limits^{\pi}_b\int\limits^1_0\int\limits^{\sqrt{1-z^2}}_0 z\sin(x)\;dy\;dz \;dx[/tex].
To know more about iterated integral:
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PLEASE HELP MUST BE AT LEAST 5 SENTENCES PLEASE HELP DUE IN 5 MINS
Answer:
The Liberty Tree (1646–1775) was a famous elm tree that stood in Boston near Boston Common, in the years before the American Revolution. The tree became a rallying point for the growing resistance to the rule of Britain over the American colonies, and the ground surrounding it became known as Liberty Hall. By hanging the noose there it is symbolizing liberty and justice is served.
Step-by-step explanation:
please mark brainliest, I hope this helps
Help me with this math question please !!!
Answer:
-1 and 14 if you put these numbers in the place of x
Please help. Thank you I am out of points because people are stealing them.
1/2
¿Cuántos envases de 1/2 de litro
se pueden llenar con el líquido de
una tina de 12 litros?
In tug of war, when one team moves toward the other, what can be said about the forces?
Answer:
When the force exerted by one team slackens, the force exerted by their third law, each team pulls the opposite team with equal force in a tug of war. So the two forces don't balance each other and the rope accelerates towards the stronger team.
Step-by-step explanation:
Pl help it’s for a grade
Answer:
A
Step-by-step explanation:
Answer:
23 m, South
Step-by-step explanation: He traveled 25 m north then he went 48 m south so that means you would have to subtract 25 from 48 to get 23 m, south
Match the event with the most appropriate probability word: The roll
a standard fair dice will show the square root of 49.
Impossible
Unlikely
Certain
Evens
Likely
Other:
Answer:
Impossible
Step-by-step explanation:
Impossible is the answer as the root of 49 is 7 and dice has numbers up till six only
Helpppop meeeee plzzzzz
Answer:
7.4
Step-by-step explanation:
solve for side a in first triangle:
[tex]10^2-6^2=y^2[/tex]
100-36=[tex]\sqrt{64}[/tex] (take square root of 64)
y=8
then use 8 to solve for x:
[tex]8^2-3^2=x^2[/tex]
64-9=[tex]\sqrt{55}[/tex] (take square root of 55)
x=7.4
Which expressions are equivalent to z ⋅ 3 1/3
select all that apply!!!!!!!!!!!!!!
3z+1/3
6z−2/3z−3z+z
z ⋅ 4−2/3
z(4−2/3)
z(1/3+3)
Answer:
Options 2), 4 , and 5 are CORRECT
Step-by-step explanation:
Notice that z * 3 1/3 = z * (3 + 1/3) = z * (9/3 + 1/3) = z * 10/3 = 10/3 z
1) The first option is not good'
2) 6 z - 2/3 z - 3 z + z = 4 z - 2/3 z = 12/3 z - 2/3 z = 10/3 z So this option is CORRECT
3) z * 4 - 2/3 is not good since 2/3 is not multiplied by z
4) z ( 4 - 2/3) = z (12/3 - 2/3) = z * 10/3 so this one is CORRECT
5) z (1 / 3 + 3) = z ( 1/3 + 9/3 ) = z (10/3) So this one is also CORRECT
Pls help extra points and mark brainlist
Answer:
80 + 30h
Step-by-step explanation:
(10 * 8) + (10 * 3h)
=80 + 30h
[tex]\Large\boxed{\tt Anwer:~80+30h}[/tex]
[tex]\tt Step-by-step~explanation:[/tex]
We need to distribute the 10 to 8 and 3h. In order to do this, we can multiply 10 by 8 and 10 by 3h.
[tex]\tt 10*8=80\\10*3h=30h[/tex]
Then, we add them together to get the answer.
[tex]\tt 80+30h[/tex]
Explain why triangles with 3 congruent parts aren’t necessarily congruent.
Answer:
SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal. If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.
Step-by-step explanation:
I hope this helps and pls mark me brainliest :)
What two numbers multiplied equal 6 but added together equal -7?
Answer:
-6 and -1
Step-by-step explanation:
let's suppose there are two numbers x and y so that it that works that would give the following equations
[tex]x*y=6 \\ x+y =-7[/tex]
now let'S rewrite the first to the form y =
[tex]y=\frac{6}{x}[/tex]
and plug that into the second one
[tex]x+\frac{x}{6} = -7|*6\\6x+x=-42\\7x=-42|:7\\x=-6[/tex]
use that to find out what y is
[tex]-6+y=-7|+6\\y=-1[/tex]
you can check that with the first equation
Calculate the following expressions :
A = 36.8 + 4 x 10.2 - (4.5 x 2): 3
Te ajut eu la dacă vrei la matematica
Does anyone know how to do this? They all are either add, subtract, multiply, divide, square, or square root besides the 4th sentence. The 4th sentence is either perfect square of a binomial, linear equation, or quadratic equation in standard form
Answer:
Square, multiply b, add, quadratic equation
Step-by-step explanation:
Michael is building a skateboard ramp He cannot decide how long the ramp should be so he draws the blueprint below. Michael draws a vertical line in the big triangle AAGH to form the AABC Michael determines the slope of both triangles to find out which ramp will be steeper
Answer:
The answer is below
Step-by-step explanation:
The slope of a line passing through two points [tex](x_1,y_1)\ and\ (x_2,y_2)[/tex] is given by:
[tex]slope(m)=\frac{y_2-y_1}{x_2-x_1}[/tex]
The slope of triangle ABC is gotten by finding the slope of line AC. Line AC passes through the point A(15, 1) and C(10, 3). The slope of triangle ABC is given by:
[tex]slope\ of\ triangle\ ABC=\frac{3-1}{10-15}=-0.4\\[/tex]
The slope of triangle AGH is gotten by finding the slope of line AH. Line AH passes through the point A(15, 1) and H(0, 7). The slope of triangle AGH is given by:
[tex]slope\ of\ triangle\ AGH=\frac{7-1}{0-15}=-0.4\\[/tex]
Since both triangles have the same slope, therefore their steepness is the same.
I need help asap!!!!
Step-by-step explanation:
step 1. a circle has an area, A, of 81(pi)
step 2. A = (pi)r^2 = 81(pi)
step 3. therefore r^2 = 81 or r = 9
step 4. if the radius is 9 inches then the diameter is 18 inches.
write a equation in the form y= mx or y= mx + b to represent the relationship between the independent and dependent qualities?
Answer:
y = 10x
Step-by-step explanation:
Independent variable (x) = time (minutes)
Dependent variable (y) = distance (feet)
The line passes through the point of origin (0, 0), which means it represents a proportional relationship. Therefore, the equation would be in the form of y = mx. Where,
m = slope = y/x
Using any point on the line, say (5, 50),
m = 50/5
m = 10
To write the equation, substitute m = 10 into y = mx
Thus:
y = 10x
plz help ASAP!!!!!!!!
Answer:
The answer is 95.03
Step-by-step explanation
The formula that I used
A=πr^2
Area = π·5.5^2 ≈ 95.03318
Hope that helps :)
Help me please thxxxxxxx
Answer: 2.4
Step-by-step explanation:
a²+b²=c²
3.9+b²=6.3
-3.9 -3.9
_________
b=2.4
check it:
3.9+2.4=6.3
there you go
mark brainly if this helps PLEASE.
1/2 to the 2nd power plus 1/2 to the 2nd power
Answer:
1/2
Step-by-step explanation:
([tex]\frac{1}{2}[/tex])² + ([tex]\frac{1}{2}[/tex])² = 1/4 + 1/4 = 2/4 or 1/2
What is the measure of angle x round the nearest degree
Answer: 16 degrees obviously
Step-by-step explanation:
On a coordinate plane, triangle A has points (negative 4, 0), (negative 4, 4), (0, 4). Triangle A prime has points (1, 0), (1, 2), (3, 2).
The pre-image, A, was dilated by a factor of 1/2 about the origin. It was then transformed in another way.
Identify the transformation after the dilation.
rotation 90° counterclockwise
rotation 180° counterclockwise
translation up 3 units
translation right 3 units
Answer:
d
Step-by-step explanation:
Answer:
that person is wrong it is c
Step-by-step explanation:
i did it
I need help pls this is to confusing
Answer: the value of a is less than b because its to the left
Step-by-step explanation:
Consider the following ordered data: 2 5 5 6 7 7 8 9 10 find Q3+1.5
MATH HELP ME NOWWWWWWWWWWWWWWWW
Answer:
part A maybe
Step-by-step explanation:
3/4+1/8 =7/8
part B
5/8 cups
PLEASE HELP PLSS IM ON A TIME LIMIT!!!
Answer:
so on the first one all your doing is multiplying 65 by the number on top so the graph should look like this.
2 3 4 5
130 195 260 325
On the second one all your doing is putting the dot where the numbers meet up so for example. I'm going to try my best at this to create a drawing so pretend ! is a line and - is also a line.
10!
!
8!
!
4!_|●_|_|_|_|_|_|
!_|_|_|_|_|_|_|_|
!_|_|_|_|_|_|_|_|_
0!_|_|_|_|_|_|_|_|_|_
2
so I hope this made sense so where I put my dot is where the blue dot should go on the graph
Please help for instant brainliest.
Answer:
(8, -1)
Step-by-step explanation:
its the point of intersection
Answer:
D
Step-by-step explanation:
the following lines meet up at the poitn (8,-1) which is to answer d
hope this helps
11.
8. The graph below shows the price Joanna
sells tomatoes according to their weight.
Write an equation for the graph.
h
H
Tomato Prices
12
10
Price (in dollars)
12
12.
Weight
(in pounds)
a. y = -x
6
10
b. y =
12
c.
=
d. y = x
Answer:
the question is invalid. please rewrite the question.
The ice cream store offers schools a discount of 12% for the month of June. The school decides to purchase ice cream bars for the end of the year picnic.
a. How much money would the school save if the school purchased 55 boxes of ice cream? Show your work and write your answer in the space below.
b. What is the total price with the discount for 55 boxes of ice cream?
PLS HELP ME!!!
Answer:
Divide the number of ice-cream bars into the number of people. Example: say you have 20 ice cream bars and 10 people you divide 20/10 and get 2 ice cream bars per person
Step-by-step explanation:
Are you also a simp for draco? :3
The distance around one block in Ava's neighborhood is 34 mile. If Ava rides her bike around the block 5 times, how many miles does she ride? Use the number line to help.
Answer:
I think the answer is 3 3/4
Step-by-step explanation:
5 × 3/4 = 15/4
15/4 = 3 3/4