It looks like the integral might be
[tex]\displaystyle \int (9 - x^2)^{3/2} \, dx[/tex]
or perhaps
[tex]\displaystyle \int \frac1{(9 - x^2)^{3/2}} \, dx[/tex]
Take note of the fact that both integrands are defined only over the interval -3 < x < 3.
For either integral, we substitute x = 3 sin(θ) and dx = 3 cos(θ) dθ.
Note that we want this substitution to be reversible, so we must restrict -π/2 ≤ θ ≤ π/2, an interval over which sine has an inverse. Then θ = arcsin(x/3).
The first case then reduces to
[tex]\displaystyle \int (9 - (3\sin(\theta))^2)^{3/2} (3 \cos(\theta) \, d\theta) = 3 \times 9^{3/2} \int (1 - \sin^2(\theta))^{3/2} \cos(\theta) \, d\theta \\\\ = 81 \int (\cos^2(\theta))^{3/2} \cos(\theta) \, d\theta \\\\ = 81 \int |\cos^3(\theta)| \cos(\theta) \, d\theta[/tex]
By definition of absolute value,
[tex]\displaystyle 81 \int |\cos^3(\theta)| \cos(\theta) \, d\theta = \begin{cases}\displaystyle 81 \int \cos^4(\theta) \, d\theta & \text{if }\cos(\theta) \ge 0 \\ \displaystyle -81 \int \cos^4(\theta) \, d\theta & \text{if }\cos(\theta) < 0\end{cases}[/tex]
and these cases correspond to 0 ≤ θ < π/2 and π/2 < θ ≤ π, respectively. But we are assuming -π/2 ≤ θ ≤ π/2, so the negative case doesn't matter to us.
You can compute the remaining antiderivative by exploiting the half-angle identity for cosine,
[tex]\cos^2(\theta) = \dfrac{1 + \cos(2\theta)}2[/tex]
Then
[tex]\cos^4(\theta) = \left(\cos^2(\theta)\right)^2 = \dfrac{1 + 2\cos(2\theta) + \cos^2(2\theta)}4 = \dfrac{3 + 4\cos(2\theta) + \cos(4\theta)}8[/tex]
and so
[tex]\displaystyle \int \cos^4(\theta) \, d\theta = \dfrac{12\theta + 8\sin(2\theta) + \sin(4\theta)}{32} + C[/tex]
We can simplify this using the double angle identity for (co)sine,
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = 1 - 2 sin²(θ)
as well as the relations,
sin(arcsin(x/3)) = x/3
cos(arcsin(x/3)) = √(9 - x²)/3
which gives us
[tex]\displaystyle \int \cos^4(\theta) \, d\theta = \dfrac{12\theta + 16 \sin(\theta) \cos(\theta) + 4 \sin(\theta) \cos(\theta) (1 - 2\sin^2(\theta))}{32} + C[/tex]
Putting this in terms of x, we get
[tex]\displaystyle \int (9 - x^2)^{3/2} \, dx \\ = 81 \times \dfrac{12\arcsin\left(\frac x3\right) + 16 \times \frac x3 \times \frac{\sqrt{9-x^2}}3 + 4\times\frac x3\times\frac{\sqrt{9-x^2}}3 \left(1 - 2\left(\frac x3\right)^2\right)}{32} + C[/tex]
[tex]\displaystyle \int (9 - x^2)^{3/2} \, dx = 81 \times \dfrac{12\arcsin\left(\frac x3\right) + \frac{16x\sqrt{9-x^2}}9 + \frac{4x(9-2x^2)\sqrt{9-x^2}}{81}}{32} + C[/tex]
[tex]\boxed{\displaystyle \int (9 - x^2)^{3/2} \, dx = \dfrac{12\arcsin\left(\frac x3\right) + (180x-8x^3)\sqrt{9-x^2}}{32} + C}[/tex]
If you were asking about the other integral, the first few steps are similar and you end up with the far more trivial integral and antiderivative
[tex]\displaystyle \frac19 \int \frac{d\theta}{\cos^2(\theta)} = \frac19 \int \sec^2(\theta) \, d\theta = \frac19 \tan(\theta) + C[/tex]
Putting it back in terms of x, we get
[tex]\displaystyle \int \frac1{(9 - x^2)^{3/2}} \, dx = \frac19 \tan\left(\arcsin\left(\frac x3\right)\right) + C[/tex]
Recall that tan(θ) = sin(θ)/cos(θ), so
[tex]\displaystyle \int \frac1{(9 - x^2)^{3/2}} \, dx = \frac19 \times \frac{\frac x3}{\frac{\sqrt{9-x^2}}3} + C[/tex]
[tex]\boxed{\displaystyle \int \frac1{(9 - x^2)^{3/2}} \, dx = \frac{x}{9\sqrt{9-x^2}} + C}[/tex]
What does PITI stand for?
who want friends
Answer:
it's stands for principal, interest, taxes, and insurance.
Step-by-step explanation:
PITI is a. acronym that stands for principal, interest, taxes, and insurance. Mortgage lenders usually estimate PITI for you before they decide whether you qualify for mortgage.
Answer:
principal, interest, taxes and insurance (if this is the one you mean
Step-by-step explanation:
PITI is an acronym that stands for principal, interest, taxes and insurance! Some mortgage lenders estimate by using PITI before they decide whether you qualify for a mortgage.
hope this helped :D
-dani
Fill in the blank to complete the trigonometric formula. (Select all that apply.)
COS 2u
Answer:
2cos^2u--1
1--2sin^2u
cos^2u--sin^2u
Step-by-step explanation:
please mark me as brainlest
3 folders cost $2.91.
Which equation will help determine the cost of 2 folders?
Answer:
5.82
Step-by-step explanation:
multyply 2 x 2.91
the 2 folders are 5.82
Answer:
B
Step-by-step explanation:
if you cross-multiply the proportion in answer B you will get:
3x = 2(2.91)
3x = 5.82
x = 5.82÷3
x = 1.94
1.94 is the cost for 2 folders if 3 cost $2.91
Solve the following inequality. 2(P + 1) > 7 + P
Answer: p > 5
Step-by-step explanation: screenshot
Answer:
p > 5
Step-by-step explanation:
2(p + 1) > 7 + p ← distribute parenthesis on left side
2p + 2 > 7 + p ( subtract p from both sides )
p + 2 > 7 ( subtract 2 from both sides )
p > 5
William buys a book for $14.85. If he pays with a $20 bill, what will his
change c be? Write and solve an equation to find out.
Answer:
$5.15
Step-by-step explanation:
$20
-$14.85 =$5.15
simplify 3 x 1/9
A. 1/3
B. 3/9
C. 28/9
D. 3 1/9
Answer:
A. 1/3
Step-by-step explanation:
3x1/9=3/9
3/9 simplified = 1/9
6.34 in expanded form
Answer:
Expanded Notation Form:
6.34
=
6
+
0.3
+
0.04
Expanded Factors Form:
6.34
=
6 ×
1
+
3 ×
0.1
+
4 ×
0.01
Expanded Exponential Form:
6.34 =
6 × 100
+
3 × 10-1
+
4 × 10-2
heeeeeeeeeeeelp dis to hard
Solution:
Note that:
y = -x + 1630 = 2x + ySubstitute the value of y into the equation to find x.
30 = 2x + y=> 30 = 2x - x + 16=> 30 = x + 16=> 30 - 16 = x=> x = 14Substitute the value of x into any equation to obtain the value of y.
30 = 2x + y=> 30 = 2(14) + y=> 30 = 28 + y=> 30 - 28 = y=> y = 2Putting the values in (x,y) form.
(x,y) ⇒ (x = 14,y = 2) ⇒ (14,2)Which table shows a proportional relationship between x and y?
4
7
8
10
O
2
3.5
4
y
5
2
х
4
7
8
O
6
у
10
21
24
х
1
2
4
3
o
00
6
8
10
12
y
40
50
60
X
90
O
8
10
14
y
18
Already answered here: https://brainly.com/question/26800677
The first table represents the proportional relationship between x and y
What is Ratio?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.
Proportional relationships are relationships between two variables where their ratios are equivalent.
In the first table the ratio of each variables are equal.
4/2 = 7/3.5 = 8/4= 10/5 =2
So the first table represents a Proportional relationship.
In the remaining tables there is no equal ratio between variables x and y.
Hence, the first table represents the proportional relationship between x and y
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Hello can someone explain to me how to do this?
Vx=o/a
a. Solve for x
b. Solve for o
C.Solve for a
The managers at smiths bbq have been asked to cut?
Answer:
huh?
I don't understand this question
solve this with an explantion
Answer:
X=1
Step-by-step explanation:
6(x+4)=30
first you do the foil method with the 6(x+4) to get 6x+24.
Your new equation is now 6x+24= 30
Now you subtract 24 from 30 to get 6.
Now your equation is 6x=6.
From there you divide 6 from 6 to get 1. Therefore x=1
there are 65 chairs set up for a movie they are arrange in 5 rows. each row has the same number of chairs
how many chairs are in each row
Answer:
13
Step-by-step explanation:
because what x 5=65 ?
? now =13
13 x 1=13/row 1
13 x 2=26/row 2
13 x 3=39/row3
13 x 4=52/row 4
13 x 5=65/row 5
hope this helps have a awesome afternoon:)
What is the range of the given function?
{x | x = -5, 1, 4, 6}
{fyly=-7, -1, 0, 9}
{x / x=-7, -5, -1, 0, 1, 4, 6, 9}
{y y=-7, -5, -1, 0, 1,4,6,9}
great questing but I don't know I just want my free answers lol
H is between Q and P
QH= 91
QP= 116
What is HP ?
Answer:
25
Step-by-step explanation:
UP = QP - QH = 116 - 97 = 25
Darren wants to build a regulation size pool table that is 9 feet in length. The plans he ordered are 18 x 36”. What is the scale factor of the dilation he must used to build a regulation size table
Answer:
Width and length of the plan is 18 inches and 36 inches. The required length of the table is 9 feet, which is 9 * 12 = 108 inches
So scale factor = [tex]\frac{Required length of the table}{Length of the plan}[/tex] = [tex]\frac{108}{36}[/tex] = 3
The answer is 3
Pls mark as brainliest
Sam had 19 kilograms of candy. If he wanted to split the candy into 2 bags, how much should be in each bag? Between what two
whole numbers does your answer lie?
Answer:
in each bag there is 9.5 kg of candy
range is between 9 and 10
Answer:
9.5 kg in each
Step-by-step explanation:
Sam has 19 kg of candy and he wants to split it into 2 equal bags. Then it is 19 divided by 2 which is 9.5 kg
a box of candy is shaped like a triangular pyramid. the volume of the candy box measures 120in cubed. the area of the base of the candy box is 18in squared. what is the height of the candy box?
Answer:
Height
A: Base Area
Volume = Ah/3
base=18
volume area of the box =120
=720
The temperature, t, in degrees Fahrenheit, can be found by counting the number of cricket chirps, c, heard in 14 seconds and then adding 40. The equation t = c + 40 models the relationship between the temperature and the number of cricket chirps.
What is true about the graph that represents this real-world scenario? Select two options.
Answer:
B & E
Step-by-step explanation:
All values of t must be positive
A viable solution is (10, 50)
got it right on edge
William has 3 shipping containers that measure 4 feet by 8 feet by 16 feet. He wants to fill all of the containers with 4-foot cubes. How many cubes can he pack if all the containers are full?
If each cube can get William $24.25 profit, how much will he make?
Answer:
i think the anwer 45
Step-by-step explanation:
because did math
William can fit 384 cubes in the containers altogether. Then you have to multiply 384 by 24.25 to find out how much he would make.
384
x 24.25
$9,312
Write the equation of the following line in slope-intercept form.
A. y=2/5 x + 4
B. y=5/2 x -10
C. y=2/5 x - 10
D. y=5/2 x +4
Answer:
y=2/5x+4 Answer: A)
Step-by-step explanation:
Translate the following sentence into an equation. Use x to represent an unknown number. Do not solve.
The product of 6 and the sum of a number and 5 results in the difference of 5 and the number.
Write an equation for the given conditions.
(Do not simplify.)
PLS HELP ME, PLS THIS IS DUE FIRST CORRECT ANSWER GETS BRAINLIEST
Answer:
c and d are your answer!
Step-by-step explanation:
Just tested on this!
Trust me!
Have an amazing day!
15. Find the length of BC. Leave your answer in
radical form.
Α.
1600
10 cm
B
с
the answer is 10 CM because the length is10 cm both all side
Find the solution(s) for x in the equation below. (x + 4)(x - 12) = 0
Answer:
-4 or 12
Step-by-step explanation:
(x + 4)(x - 12) = 0
so
x + 4 = 0 OR x - 12 = 0
x + 4 = 0 leads x = -4
x - 12 = 0 leads x = 12
Answer:
x = -4 or x = 12
Step-by-step explanation:
(x + 4)(x - 12) = 0 means:
(x + 4) = 0 or (x - 12) = 0
x + 4 = 0
Subtract 4 from both sides:
⇒ x + 4 - 4 = 0 - 4
⇒ x = -4
x - 12 = 0
add 12 to both sides:
⇒ x - 12 + 12 = 0 + 12
⇒ x = 12
Therefore the solutions are x = -4 or x = 12
4. Circle the number sentences that are true.
Cross out the number sentences that are false 30> 10 30 <10 10>30 10 <30
Answer:
Step-by-step explanation:
True statements :
30 > 10
10 < 30
False statements :
30 < 10
10 > 30
Circle: 30 > 10, 10 < 30
Cross out: 30 < 10, 10 > 30
What is inequality?
It shows a relationship between two numbers or two expressions.
There are commonly used four inequalities:
Less than = <
Greater than = >
Less than and equal = ≤
Greater than and equal = ≥
We have,
We are given four number inequalities:
30 > 10
30 < 10
10 > 30
10 < 30
The first number sentence, "30 > 10", is true, since 30 is greater than 10.
The second number sentence, "30 < 10", is false, since 30 is not less than 10.
The third number sentence, "10 > 30", is false, since 10 is not greater than 30.
The fourth number sentence, "10 < 30", is true, since 10 is less than 30.
Therefore,
We would circle the first and fourth number sentences as true, and cross out the second and third number sentences as false.
Learn more about inequalities here:
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Help please I beg of you
Answer:
45.7 meters
Step-by-step explanation:
15 = 6 + 7y
9 = 7y
y = 1.3
The two lengths are still the same, so you can set the two parts equal to the whole. Now that you have y you can substitute it in and add all the sides together.
3y + 7y + 3y + 6y + 6 +15 = P
3(1.3) + 7(1.3) + 3(1.3) + 6(1.3) + 6 + 15 = P
P = 45.7 meters
What is the slope and y-intercept? 20 points!!
Answer:
Slope = 2
Y-intercept = -4
Step-by-step explanation:
The slope of a function is found by finding the change in the y-values and dividing it in the change in x-values.
The change in y-values is 2 because that is the difference between each given data point. The change in x-values is 1 for the same reason.
The slope would be 2/1 which is just 2.
The y-intercept can be found by setting up the function in slope-intercept form (y=mx+b) and using the given data to solve for b. The point I chose to do this with was (2,0)
y=mx+b
y=2x+b
0=(2*2)+b
0=4+b
b=-4
So the slope is 2 and the y-intercept is -4.
PLEASE HELP ME!! what is the answer?
Answer:
A is the answer
Step-by-step explanation:
I hope it helps
24 is 120% of what?
Answer:
[tex]\frac{24}{20}= \frac{120}{100}[/tex]
Step-by-step explanation:
120/100 * x = 24
1.2 * x = 24
x = 20
24 is 120% of 20
The answer missing blanks to your problems are [tex]\frac{24}{20}= \frac{120}{100} \\[/tex]
Hope this helps :)
Have a nice day!
