-50 points- matrix system

Answer:

Evaluate 8P6 P 6 8 using the formula nPr=n!(n−r)! P r n = n ! ( n - r ) ! . 8!(8−6)! 8 ! ( 8 - 6 ) ! Subtract 6 6 from 8 8 . 8!(2)! 8 ! ( 2 ) ! Simplify 8!(2)! 8 !

Step-by-step explanation:

evaluate" usually means to put a value in for the variable, but you don't give us a value for p. also, it is unclear if you ...

The value of the expression [tex]^8P_6[/tex] is 20160.

A permutation of a set in mathematics is, broadly speaking, the rearrangement of its elements if the set already has an ordered structure into a sequence or linear order.

The value of the expression is calculated as:-

[tex]^8P_6=\dfrac{8!}{8!-6!}=\dfrac{8!}{2!}[/tex]

[tex]^8P_6 =\dfrac{8\times 7\times 6\times 5\times 4\times 3\times 2}{2}[/tex]

[tex]^8P_6[/tex] = 20160

Hence, the value is 20160.

To know more about permutations follow

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Answer:

y = -4x - 6.

Step-by-step explanation:

We are given (-5, -3), (-1, -2), and (3, -1) for points of a line. First, we need to find the slope.

(-2 - -3) / (-1 - -5) = (-2 + 3) / (-1 + 5) = 1 / 4.

A perpendicular bisector would have a slope of -4, which is the negative reciprocal of 1/4.

Now that we have the slope, we can say that the equation is y = -4x + b. To find what is b, we can say that y = -2 and x = -1.

-2 = -4(-1) + b

-2 = 4 + b

b + 4 = -2

b = -6

So, the equation of the perpendicular bisector is y = -4x - 6.

Hope this helps!

Answer:

y = -4x - 6.

Step-by-step explanation:

Just took the test and got it right

Answer:

15.8 sq. in. of paper will be required.

Step-by-step explanation:

The problem is that a drinking cup does not have a cover, so only the lateral surface area counts.

I.e. We need only the first term.

A = pi r l = pi * 1.5 * sqrt(3^2+1.5^2)

= 15.81 sq. in.

Answer:

The first picture's answer would be (6, 21)

Step-by-step explanation:

You have to find the points on the 8th and the 9th day, and then you would add them together, and then divide by two finding the average, which would be 24 and 18, so when added, you get 42, divided by 2 you get 21. You look on the graph for the point with 21, and you find it is on 6.

Answer:

r = 0.023104906

Step-by-step explanation:

Given half life = T = 30 yrs.

Decay constant = r.

Using the decay constant formula:

[tex]r=\frac{\ln2}{T}\\r=\frac{\ln2}{30}\\r=0.023104906[/tex]

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Answer:

[tex]r=\frac{\sqrt{3} }{4}[/tex]    and    [tex]r=-\frac{\sqrt{3} }{4}[/tex]

Step-by-step explanation:

when you solve for r in the given equation, you need to apply the square root property, which gives positive and negative answers (both should therefore be considered):

[tex]r^2=\frac{3}{16} \\r=+/-\sqrt{\frac{3}{16}} \\r=+/-\frac{\sqrt{3} }{4}[/tex]

then you need to include these two possible solutions:

[tex]r=\frac{\sqrt{3} }{4}[/tex]    and    [tex]r=-\frac{\sqrt{3} }{4}[/tex]

Answer:

A: y= 1/4x - 7

if it is perpendicular, then it creates 4 right angles. so that new line would pass through (0,-7) and something else that isnt important. but the slope, or m, would be 1/4, and the y intercept would be -7. so the new equation is y=1/4x-7

Answer:

3* 10 ^12

Step-by-step explanation:

(5x10^7)(6x10^4)

Multiply the numbers together

5*6 =30

Add the exponents

10^7 * 10 ^ 4 = 10 ^(7+4) = 10 ^11

30 * 10 ^11

But this is not scientific notation since 30 >10

Move the decimal one place to the left and add 1 to the exponent

3* 10 ^12

Answer:

3* 10 ^12

Step-by-step explanation:

Answer:

x=nπ3, n∈I

Step-by-step explanation:

sin x + sin 5x = sin 2x + sin 4x

⇒⇒   2 sin 3x cos 2x = 2 sin 3x cos x

⇒⇒   2 sin 3x(cos 2x - cos x) = 0

⇒    sin 3x=0 ⇒ 3x=nπ ⇒ x=nπ3⇒    sin 3x=0 ⇒ 3x=nπ ⇒ x=nπ3 , n∈I, n∈I

or    cos 2x−cos x=0 ⇒ cos 2x=cos xcos 2x-cos x=0 ⇒ cos 2x=cos x

⇒   2x=2nπ±x  ⇒  x=2nπ, 2nπ3⇒   2x=2nπ±x  ⇒  x=2nπ, 2nπ3 , n∈I, n∈I

But solutions obtained by x=2nπx=2nπ , n∈I, n∈I or x=2nπ3x=2nπ3 , n∈I, n∈I are all involved in x=nπ3x=nπ3 , n∈I

Answer:

We accept H₀  data from the survey is not enough to claim that 50% of the proportion indicated in previous studies have change

Step-by-step explanation:

To get conclusions about the survey we need to develop a hypothesis test of proportion

According to previous studies, (p₀ ) 50 % of staff and customers use public transportation, and we got from a survey 0f 1002 people 483 responded they also use then  p = 483/1002 then

n sample size is   1002  and  p = 0,482    (48,2 % )

Test Hypothesis

Null hypothesis                                   H₀            p  =  p₀

Alternative hypothesis                       Hₐ            p  <  p₀

CI =  95 %    α  = 5 %     α = 0,05    and from z-table we find z score for that value    z(c)  =  - 1,64

z(s)  =  (  p  -  p₀ ) /  √ (p₀*q₀)/ n         p₀ = q₀  = 0,5

z(s)  =  - 0,018* 31,65 / 0,5

z(s)  =  - 1,1394

To compare

z(s)  and  z(c)          -1,1394 > 1,64

Then z(s) is inside the acceptance region. We accept H₀ , because we don´t have enough evidence to claim that the survey results indicate a change in

the original proportion

Answer:

The margin of error is  [tex]E = 1.96 * \frac{ 0.92}{\sqrt{28 } }[/tex]

Step-by-step explanation:

From the question we are told that

    The sample size is  [tex]n = 28[/tex]

     The  sample  mean is  [tex]\= x = 2.4 \ hr[/tex]

      The  standard deviation is  [tex]\sigma = 0.92 \ hr[/tex]

Given that the confidence level is 95% the the level of significance can be evaluated as

             [tex]\alpha = 100 -95[/tex]

            [tex]\alpha = 5 \%[/tex]

             [tex]\alpha = 0.05[/tex]

Next we obtain the critical value of  [tex]\frac{\alpha }{2}[/tex] from the normal distribution table,the value is  [tex]Z_{\frac{\alpha }{2} } = Z_{\frac{0.05}{2} } = 1.96[/tex]

Generally the margin of error is mathematically represented as

           [tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }[/tex]

substituting values

          [tex]E = 1.96 * \frac{ 0.92}{\sqrt{28 } }[/tex]

         [tex]E = 0.3408[/tex]

Answer:

  35 cm

Step-by-step explanation:

The volume of a cylinder is given by ...

  V = πr²h

We want to find h for the given volume and diameter. First, we must convert the given values to compatible units.

  1 L = 1000 cm³, so 99 L = 99,000 cm³

  60 cm diameter = 2 × 30 cm radius

So, we have ...

  99,000 cm³ = π(30 cm)²h

  99,000/(900π) cm = h ≈ 35.01 cm

The oil is 35 cm deep in the drum.

Answer:

6.02

six point zero two

Step-by-step explanation:

Answer:

602 / 100= 6,02

Step-by-step explanation:

602 to divide 100 = 6,02

the graph has 12 segments so angle enclosed by each segment is [tex] {2\pi\over 12}=\frac{\pi}6[/tex]

anti-clockwise is taken as positive, so if you want positive q, you need to rotate 8 segments [tex] q=8\frac,{\pi}6=\frac{4\pi}3 [/tex] , and and 8 circles or units so r=8

and for a negative angle, you need to rotate clockwise

Which is 4 segments from the horizontal line. so [tex]q=-\frac{2\pi}3[/tex] and r will be same, 8 units.

[not sure about -r so I won't include it in answer]

Answer:

Points : ( 8, - 2π/3 ), ( - 8, π/3 ), ( - 8, - 5π/3 )

Step-by-step explanation:

For the first two cases, ( r, θ ) r would be > 0, where r is the directed distance from the pole, and theta is the directed angle from the positive x - axis.

So when r is positive, we can tell that this point is 8 units from the pole, so r is going to be 8 in either case,

( 8, 240° ) - because r is positive, theta would have to be an angle with which it's terminal side passes through this point. As you can see that would be 2 / 3rd of 90 degrees more than a 180 degree angle,or 60 + 180 = 240 degrees.

( 8, - 120° ) - now theta will be the negative side of 360 - 240, or in other words - 120

Now let's consider the second two cases, where r is < 0. Of course the point will still be 8 units from the pole. Again for r < 0 the point will lay on the ray pointing in the opposite direction of the terminal side of theta.

( - 8, 60° ) - theta will now be 2 / 3rd of 90 degrees, or 60 degrees, for - r. Respectively the remaining degrees will be negative, 360 - 60 = 300, - 300. Thus our second point for - r will be ( - 8, - 300° )

_________________________________

So we have the points ( 8, 240° ), ( 8, - 120° ), ( - 8, 60° ), and ( - 8, - 300° ). However we only want 3 cases, so we have points ( 8, - 120° ), ( - 8, 60° ), and ( - 8, - 300° ). Let's convert the degrees into radians,

Points : ( 8, - 2π/3 ), ( - 8, π/3 ), ( - 8, - 5π/3 )

Answer:

87; 2%

Step-by-step explanation:

An exponential growth model is defined as :

F(x) = A( 1 + r)^x

Where;

A = Initial amount,

r = rate of increase

x = time

Comparing the exponential growth function with the exponential growth model given;

f(x)=87(1.02)^x

A = 87 = Initial amount

The growth rate of the model expressed as a percentage :

Taking :

(1 + r) = 1.02

1 + r = 1.02

r = 1.02 - 1

r = 0.02

Expressing r as a percentage :

0.02 * 100% = 2%

Answer: a = 9, b = 48, c = -1

Step-by-step explanation:

"a" represents the points you receive if an Ace is picked.  It is given that you get 9 points ----> a = 9

"b" represents the number of cards that are Not an Ace.  4 cards in the deck are Aces so 52 - 4 = 48 cards are Not an Ace -----> b = 48

"c" represents the points you receive if Not an Ace is picked.  It is given that you lose 1 point ----> c = -1

Answer:

Here is the rest of the page

Step-by-step explanation:

Answer: [tex]A=6000(1.012)^t[/tex]

Step-by-step explanation:

General exponential function:

[tex]A=P(1+r)^t[/tex]

, where P= current  population

r= rate of growth

t= time period

A= population after t years

As per given , we have P=6,000

r= 1.2% = 0.012

Then, the required exponential function:  [tex]A=6000(1+0.012)^t[/tex]

or [tex]A=6000(1.012)^t[/tex]

(a) Take the Laplace transform of both sides:

[tex]2y''(t)+ty'(t)-2y(t)=14[/tex]

[tex]\implies 2(s^2Y(s)-sy(0)-y'(0))-(Y(s)+sY'(s))-2Y(s)=\dfrac{14}s[/tex]

where the transform of [tex]ty'(t)[/tex] comes from

[tex]L[ty'(t)]=-(L[y'(t)])'=-(sY(s)-y(0))'=-Y(s)-sY'(s)[/tex]

This yields the linear ODE,

[tex]-sY'(s)+(2s^2-3)Y(s)=\dfrac{14}s[/tex]

Divides both sides by [tex]-s[/tex]:

[tex]Y'(s)+\dfrac{3-2s^2}sY(s)=-\dfrac{14}{s^2}[/tex]

Find the integrating factor:

[tex]\displaystyle\int\frac{3-2s^2}s\,\mathrm ds=3\ln|s|-s^2+C[/tex]

Multiply both sides of the ODE by [tex]e^{3\ln|s|-s^2}=s^3e^{-s^2}[/tex]:

[tex]s^3e^{-s^2}Y'(s)+(3s^2-2s^4)e^{-s^2}Y(s)=-14se^{-s^2}[/tex]

The left side condenses into the derivative of a product:

[tex]\left(s^3e^{-s^2}Y(s)\right)'=-14se^{-s^2}[/tex]

Integrate both sides and solve for [tex]Y(s)[/tex]:

[tex]s^3e^{-s^2}Y(s)=7e^{-s^2}+C[/tex]

[tex]Y(s)=\dfrac{7+Ce^{s^2}}{s^3}[/tex]

(b) Taking the inverse transform of both sides gives

[tex]y(t)=\dfrac{7t^2}2+C\,L^{-1}\left[\dfrac{e^{s^2}}{s^3}\right][/tex]

I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that [tex]\frac{7t^2}2[/tex] is one solution to the original ODE.

[tex]y(t)=\dfrac{7t^2}2\implies y'(t)=7t\implies y''(t)=7[/tex]

Substitute these into the ODE to see everything checks out:

[tex]2\cdot7+t\cdot7t-2\cdot\dfrac{7t^2}2=14[/tex]

Answer:

[tex] y = - 2 [/tex]

Step-by-step explanation:

Given that the 2 triangles are congruent based on the Hypotenuse-leg theorem, this implies that:

[tex] x - y = x + 2 [/tex] , and [tex] 2x - y = 4x + 2y [/tex]

Using the expression, [tex] x - y = x + 2 [/tex], solve for y:

[tex] x - y - x = x + 2 - x [/tex]

[tex] - y = 2 [/tex]

[tex] y = - 2 [/tex]

Answer:

Step-by-step explanation:

(f/g) = (3 - 2x ) / (1/x + 5) You could go to the trouble to simplify all of this, but the easiest way is to just put in the 8 where you see an x

(f/g)8 = (3 - 2*8) / (1/8 + 5)

(f/g)/8 = (3 - 16 / (5 1/8)          1/8 = 0.125

(f/g) 8 = - 13 / ( 5.125)

(f/g)8 = - 2.54

Answer:

see below

Step-by-step explanation:

Scientific notation is a * 10 ^b

a must be a number between 1 ( including 1 ) and less than 10

12 is greater than 10 so it is not scientific notation

Answer:

the first  partial sum  [tex]\mathbf{S_1= \dfrac{1}{2}}[/tex]

the second partial sum  [tex]\mathbf{S_2= \dfrac{3}{4} }[/tex]

the third partial sum [tex]\mathbf{S_3= \dfrac{7}{8}}[/tex]

the fourth  partial sum [tex]\mathbf{S_4= \dfrac{15}{16}}[/tex]

the fifth partial sum [tex]\mathbf{S_5= \dfrac{31}{32}}[/tex]

the sixth partial sum [tex]\mathbf{S_6= \dfrac{63}{64}}[/tex]

Step-by-step explanation:

The term of the sequence are given as : [tex]\dfrac{1}{2}[/tex], [tex]\dfrac{1}{2^2}[/tex], [tex]\dfrac{1}{2^3}[/tex], [tex]\dfrac{1}{2^4 }[/tex] ,  .  .  .  

The nth term for this sequence is , [tex]\mathtt{a_n =( \dfrac{1}{2})^n}[/tex]

The nth partial sum of the sequence for [tex]\mathtt{a_1,a_2,a_3.... a_n}[/tex] is [tex]\mathtt{S_n}[/tex]

where;

[tex]\mathtt{S_n = a_1 +a_2+a_3+ ...+a_n}[/tex]

The first partial sum  is:  [tex]\mathtt{S_1= a_1}[/tex]

[tex]\mathtt{S_1= (\dfrac{1}{2})^1}[/tex]

[tex]\mathbf{S_1= \dfrac{1}{2}}[/tex]

Therefore, the first  partial sum  [tex]\mathbf{S_1= \dfrac{1}{2}}[/tex]

The second partial sum is: [tex]\mathtt{S_2= a_1+a_2}[/tex]

[tex]\mathtt{S_2= (\dfrac{1}{2})^1 + (\dfrac{1}{2})^2}[/tex]

[tex]\mathtt{S_2= \dfrac{1}{2} + \dfrac{1}{4}}[/tex]

[tex]\mathtt{S_2= \dfrac{2+1}{4} }[/tex]

[tex]\mathbf{S_2= \dfrac{3}{4} }[/tex]

Therefore, the second partial sum  [tex]\mathbf{S_2= \dfrac{3}{4} }[/tex]

The third partial sum is : [tex]\mathtt{S_3= a_1+a_2+a_3}[/tex]

[tex]\mathtt{S_3= (\dfrac{1}{2})^1 + (\dfrac{1}{2})^2+(\dfrac{1}{2})^3 }[/tex]

[tex]\mathtt{S_3= \dfrac{1}{2} + \dfrac{1}{4}+\dfrac{1}{8}}[/tex]

[tex]\mathtt{S_3= \dfrac{4+2+1}{8}}[/tex]

[tex]\mathbf{S_3= \dfrac{7}{8}}[/tex]

Therefore, the third partial sum [tex]\mathbf{S_3= \dfrac{7}{8}}[/tex]

The fourth partial sum : [tex]\mathtt{S_4= a_1+a_2+a_3+a_4}[/tex]

[tex]\mathtt{S_4= (\dfrac{1}{2})^1 + (\dfrac{1}{2})^2+(\dfrac{1}{2})^3+(\dfrac{1}{2})^4 }[/tex]

[tex]\mathtt{S_4= \dfrac{1}{2} + \dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}}[/tex]

[tex]\mathtt{S_4= \dfrac{8+4+2+1}{16}}[/tex]

[tex]\mathbf{S_4= \dfrac{15}{16}}[/tex]

Therefore, the fourth  partial sum [tex]\mathbf{S_4= \dfrac{15}{16}}[/tex]

The fifth partial sum : [tex]\mathtt{S_5= a_1+a_2+a_3+a_4+a_5}[/tex]

[tex]\mathtt{S_5= (\dfrac{1}{2})^1 + (\dfrac{1}{2})^2+(\dfrac{1}{2})^3+(\dfrac{1}{2})^4 +(\dfrac{1}{2})^5 }[/tex]

[tex]\mathtt{S_5= \dfrac{1}{2} + \dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}}[/tex]

[tex]\mathtt{S_5= \dfrac{16+8+4+2+1}{32}}[/tex]

[tex]\mathbf{S_5= \dfrac{31}{32}}[/tex]

Therefore, the fifth partial sum [tex]\mathbf{S_5= \dfrac{31}{32}}[/tex]

The sixth partial sum: [tex]\mathtt{S_5= a_1+a_2+a_3+a_4+a_5+a_6}[/tex]

[tex]\mathtt{S_6= (\dfrac{1}{2})^1 + (\dfrac{1}{2})^2+(\dfrac{1}{2})^3+(\dfrac{1}{2})^4 +(\dfrac{1}{2})^5 +(\dfrac{1}{2})^6 }[/tex]

[tex]\mathtt{S_6= \dfrac{1}{2} + \dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64} }[/tex]

[tex]\mathtt{S_6= \dfrac{32+16+8+4+2+1}{64}}[/tex]

[tex]\mathbf{S_6= \dfrac{63}{64}}[/tex]

Therefore, the sixth partial sum [tex]\mathbf{S_6= \dfrac{63}{64}}[/tex]

Answer:

[tex] y = \frac{3}{4}x - 2 [/tex]

Step-by-step explanation:

Equation of a line is given as [tex] y = mx + b [/tex]

Where,

m = slope of the line = [tex] \frac{y_2 - y_1}{x_2 - x_1} [/tex]

b = y-intercept, which is the value at the point where the line intercepts the y-axis. At this point, x = 0.

Let's find m and b to derive the equation for the line.

[tex] m = \frac{y_2 - y_1}{x_2 - x_1} [/tex]

Use the coordinate pair of any two points on the line. Let's use the following,

[tex] (0, -2) = (x_1, y_1) [/tex] => on the line, when x = 0, y = -2

[tex] (4, 1) = (x_2, y_2) [/tex] => on the line, when x = 4, y = 1

Plug in the values and solve for m

[tex] m = \frac{1 - (-2)}{4 - 0} [/tex]

[tex] m = \frac{1 + 2}{4} [/tex]

[tex] m = \frac{3}{4} [/tex]

b = -2 (the line intercepts the y-axis at this point)

Our equation would be =>

[tex] y = mx + b [/tex]

[tex] y = \frac{3}{4}x + (-2) [/tex]

[tex] y = \frac{3}{4}x - 2 [/tex]

Answer:

17/16 OR [tex]1\frac{1}{16}[/tex] minutes

Step-by-step explanation:

Since Jayla spent 1/16 of a minute AND one whole minute watching a millipede crawl, we'd need to first add the two numbers.

Since the given minute is out of 16, we can convert the one minute to 16/16.  This means we can add the other 1/16 of a minute.

This leaves us with Jayla watching the millipede for 17/16 OR [tex]1\frac{1}{16}[/tex] minutes.

Hope this helps!! <3 :)

Answer:

x and y can have many values

Step-by-step explanation:

-24x - 12y = -16

Then: 24x + 12y = 16

We know: 6x + 3y = 4

X and Y can have a lot of valoues.

6x + 3y = 4

3 ( 2x + y) = 4

2x + y= 4/3

2x+y= 1.333...

Answer:

ST, RS, RT

Step-by-step explanation:

Angles of a triangle add up to 180°.

2x + 11° + 3x + 23° + x + 42° = 180°

6x + 76° = 180°

x = 17⅓

m∠R = 2x+11° = 45⅔°

m∠S = 3x+23° = 75°

m∠T = x+42° = 59⅓°

The shortest side is opposite the smallest angle, and the longest side is opposite the largest angle.

ST, RS, RT

Classica aplicação de regra de 3:

é dito que: 1 milha = 1,6km

Logo, eis a regra de 3:

  milha           km

     1   --------    1,6

     X  --------   240

1,6X = 240.1

X = 240/1,6

Logo 240km equivalem a 150milhas

Answer and Step-by-step explanation:

This is a complete question

Trials in an experiment with a polygraph include 97 results that include 23 cases of wrong results and 74 cases of correct results. Use a 0.01 significance level to test the claim that such polygraph results are correct less than 80​% of the time. Identify the null​hypothesis, alternative​ hypothesis, test​ statistic, P-value, conclusion about the null​ hypothesis, and final conclusion that addresses the original claim. Use the​ P-value method. Use the normal distribution as an approximation of the binomial distribution.

The computation is shown below:

The null and alternative hypothesis is

[tex]H_0 : p = 0.80[/tex]

[tex]Ha : p < 0.80[/tex]

[tex]\hat p = \frac{x}{ n} \\\\= \frac{74}{97}[/tex]

= 0.7629

Now Test statistic = z

[tex]= \hat p - P0 / [\sqrtP0 \times (1 - P0 ) / n][/tex]

[tex]= 0.7629 - 0.80 / [\sqrt(0.80 \times 0.20) / 97][/tex]

= -0.91

Now

P-value = 0.1804

[tex]\alpha = 0.01[/tex]

[tex]P-value > \alpha[/tex]

So, it is Fail to reject the null hypothesis.

There is ample evidence to demonstrate that less than 80 percent of the time reports that these polygraph findings are accurate.

Answer:

claire has to leave at 3:50 from her house.

Answer:

She needs to leave by 3:50 to get there on time.

Step-by-step explanation:

4:15 - 0:25 = 3:50.

Answer:

[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV = \frac{\pi (17e^5 - 2)}{2}[/tex]

General Formulas and Concepts:
Calculus

Integration

Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method [Integration by Parts]:
[tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

Multivariable Calculus

Triple Integrals

Cylindrical Coordinate Conversions:

Spherical Coordinate Conversions:

Integral Conversion [Spherical Coordinates]:
[tex]\displaystyle \iiint_T {f( \rho, \phi, \theta )} \, dV = \iiint_T {\rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta[/tex]

Step-by-step explanation:

*Note:

Recall that φ is bounded by 0 ≤ φ ≤ 0.5π from the z-axis to the x-axis.

I will not show/explain any intermediate calculus steps as there isn't enough space.

Step 1: Define

Identify given.

[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV[/tex]

[tex]\displaystyle \text{Region E:} \ x^2 + y^2 + z^2 = 25 \ \text{bounded by first octant}[/tex]

Step 2: Integrate Pt. 1

Find ρ bounds.

Find θ bounds.

Find φ bounds.

Step 3: Integrate Pt. 2

We evaluate this spherical integral by using the integration rules, properties, and methods listed above:

[tex]\displaystyle \begin{aligned} \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 \int\limits^5_0 {e^{\rho} \rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta \\ & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 {\bigg[ (\rho^2 - 2 \rho + 2) e^{\rho} \sin \phi \bigg] \bigg| \limits^{\rho = 5}_{\rho = 0}} \, d\phi \, d\theta\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 {(17e^5 - 2) \sin \phi} \, d\phi \, d\theta \\& = \int\limits^{\frac{\pi}{2}}_0 {\bigg[ -(17e^5 - 2) \cos \phi \bigg] \bigg| \limits^{\phi = \frac{\pi}{2}}_{\phi = 0}} \, d\theta \\& = \int\limits^{\frac{\pi}{2}}_0 {17e^5 - 2} \, d\theta \\& = (17e^5 - 2) \theta \bigg| \limits^{\theta = \frac{\pi}{2}}_{\theta = 0} \\& = \frac{\pi (17e^5 - 2)}{2}\end{aligned}[/tex]

∴ the given integral equals [tex]\displaystyle \bold{\frac{\pi (17e^5 - 2)}{2}}[/tex].

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Learn more about spherical coordinates: https://brainly.com/question/16415822

Learn more about multivariable calculus: https://brainly.com/question/4746216

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Topic: Multivariable Calculus

Unit: Triple Integrals Applications