Integration
IntegralsIntegration TechniquesIntegration 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]
Double Integrals
Polar Coordinates Conversions:
[tex]\displaystyle x = r \cos \theta[/tex][tex]\displaystyle y = r \sin \theta[/tex][tex]\displaystyle x^2 + y^2 = r^2[/tex]Integral Conversion [Polar Coordinates]:
[tex]\displaystyle \iint_T {f(x, y)} \, dA = \iint_R {f(r, \theta)r} \, dr \, d\theta[/tex]
The formatting of the question was thrown off, so I have defined it down below.
We are given an integral and asked to convert to polar coordinates as well as evaluate it:
[tex]\displaystyle \int \limits^{8}_{-8} \int \limits^{\sqrt{64 - x^2}}_{0} \sin(x^2 + y^2) \, dy \, dx[/tex]
It would be quite difficult to evaluate the given integral using conventional methods, so we apply polar conversion to evaluate the integral. Let's start out by converting the function and the bounds.
[Bounds] Cartesian to Polar:
[tex]\displaystyle \left \{ {{-8 \leq x \leq 8} \atop {0 \leq y \leq \sqrt{64 - x^2}}} \right \longrightarrow \left \{ {{0 \leq r \leq 8} \atop {0 \leq \theta \leq \pi}} \right[/tex]
[Function] Cartesian to Polar:
[tex]\displaystyle f(x ,\ y) = \sin(x^2 + y^2) \longrightarrow f(r ,\ \theta) = \sin r^2[/tex]
Now that we've converted to polar coordinates, we can convert the integral using our integral conversion listed under "Multivariable Calculus":
[tex]\displaystyle \int \limits^{8}_{-8} \int \limits^{\sqrt{64 - x^2}}_{0} \sin(x^2 + y^2) \, dy \, dx \longrightarrow \int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta[/tex]
We can now evaluate the polar integral using basic integration techniques listed under "Calculus":
[tex]\displaystyle \begin{aligned}\int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta & = \int \limits^{\pi}_{0} \underbrace{\int \limits^{8}_{0} r \sin r^2 \, dr \, }_{u = r^2 ,\ du = 2r \, dr} d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \int \limits^{8}_{0} 2r \sin r^2 \, dr \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \int \limits^{r = 8}_{r = 0} \sin u \, du \, d\theta \\\end{aligned}[/tex]
[tex]\displaystyle \begin{aligned}\int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta & = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( - \cos u \bigg) \bigg| \limits^{r = 8}_{r = 0} \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( - \cos r^2 \bigg) \bigg| \limits^{r = 8}_{r = 0} \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( - \cos 64 + 1 \bigg) \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( 1 - \cos 64 \bigg) \, d\theta \\\end{aligned}[/tex]
[tex]\displaystyle \begin{aligned}\int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta & = \frac{1}{2} \bigg(1 - \cos 64 \bigg) \bigg( \theta \bigg) \bigg| \limits^{\pi}_{0} \\& = \boxed{ \frac{\pi}{2} \bigg( 1 - \cos 64 \bigg) }\end{aligned}[/tex]
∴ the integral equals:
[tex]\displaystyle \boxed{ \frac{\pi}{2} \bigg( 1 - \cos 64 \bigg) }[/tex]
[tex]\displaystyle \int \limits^{8}_{-8} \int \limits^{\sqrt{64 - x^2}}_{0} \sin(x^2 + y^2) \, dy \, dx = \boxed{ \frac{\pi}{2} \bigg( 1 - \cos 64 \bigg) }[/tex]
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Topic: Multivariable Calculus
Unit: Double Integrals
find the sum and difference of the greatest and the least six digit number formed by the digits 2,0,4,7,6,5 using each digit only once
Answer:
See below
Step-by-step explanation:
The greatest number formed by those numbers is: 765420
The least number formed by those number is: 204567
The sum is: 765420 + 204567 = 969987
The difference is: 765420 - 204567 = 560853
Answer: Hi!
First we need to form the biggest and smallest number out of these digits. To do this it's pretty simple - just order the numbers from greatest to least for the greatest number, and from least to greatest for the least number! (However, for the least number, we will have to put the zero in front of 2, the second least number, because we can't put 0 in front of 2 and still be actually using it in the number.)
Greatest: 765420
Least: 204567
Now we add the numbers together:
765420 + 204567 = 969987
Now we subtract the numbers:
765420 - 204567 = 560853
These are your answers!
Hope this helps!
Using the order of operations, what should be done first to evaluate (-2)³ + 4 ÷ (-15 + 9)(3) - 4?
Answer:
use BODMAS
brackets
of
division
multiplication
addition
subtraction
or Americans would use BEDMAS I think
so it'll be 8 + 4 ÷ (-24) × (3) - 4 = 35
u can break the brackets down further
Answer:
Its a on ed
Step-by-step explanation:
When two straight lines cross, it is found that the angles opposite to each ate the same size. They are known as
The angles don't need to be lined up vertically, or have one over top another. You could have them laid out horizontally. All that matters is that they are opposite angles after an X shape has been formed by two crossing lines. Vertical angles are always congruent.
Katie is making chocolate candies for her classmates' Valentine's Day gifts. She melts 12.8 ounces of chocolate and divides it among 40 candy molds. If Katie divided the melted chocolate evenly, how much would each candy weigh?
Answer:
0.32 ounces of chocolate per mold
Step-by-step explanation:
Answer:
the correct answer is 3.125
Step-by-step explanation:
ANSWER IF YOU LOVE YOUR LORD!!!! Which graph represents the solution to: 2k+8<5k-1?
Answer:
Third graph
Step-by-step explanation:
First, find the solution to the equation of inequality given.
2k + 8 < 5k - 1
Subtract 5k from both sides
2k + 8 - 5k < 5k - 1 - 5k
-3k + 8 < -1
Subtract 8 from both sides
-3k + 8 - 8 < -1 - 8
-3k < -9
Divide both sides by -3. (Note: < will change to > when dividing both sides with negative number)
[tex] \frac{-3k}{-3} [/tex] > [tex] \frac{-9}{-3} [/tex]
k > 3
The graph that will represent this solution will show that all values of k are greater than 3. 3 is not included as a solution. The "o" on top of the 3 on the number line won't be shaded to indicate that 3 is not included. And also, the arrow will point from 3 towards our right.
Therefore, the 3rd graph is the answer.
3. Solve: 4x + 2 = 3x -9
Answer:
x=-11 is the answer
Step-by-step explanation:
4x + 2 = 3x -9
4x-3x+2=-9
1x=-9-2
x=-11
hope it will help ^_^
Phillip deposited $7,775 into a savings account 14 years ago. The account has an interest rate of 4.5% and the balance is currently $14,398.87. How often does the interest compound? (2 points)
daily
monthly
quarterly
annually
Answer:
annually
Step-by-step explanation:
Compound interest formula:
A = P(1 + r/n)^(nt)
amount of interest = balance - principal
Try daily:
A = 7775(1 + 0.045/365)^(365 * 14)
A = 14,597.88
This is too much money, so it was compounded less than daily.
Try monthly:
A = 7775(1 + 0.045/12)^(12 * 14)
A = 14,581.23
This is too much money, so it was compounded less than monthly.
Try quarterly:
A = 7775(1 + 0.045/4)^(4 * 14)
A = 14,547.16
This is too much money, so it was compounded less than quarterly.
Try annually:
A = 7775(1 + 0.045/1)^(1 * 14)
A = 14,398.87
This is the correct amount, so it was compounded annually.
Answer: annually
The sum of Micah’s age and Aria’s age is 29. Aria is 5 years older than twice Micah’s age. Let x represent Micah’s age. The equation that can be used to find Micah’s age is x + (x + 5) = 29. x + (2 x + 5) = 29. x + (2 x) = 29. x + (2 x minus 5) = 29
Answer:
B, x + (2x + 5) = 29
Step-by-step explanation:
x, being Micahs age.
the 2, as twice the age.
5, the additional amount older.
Answer:
Yup hes right
Step-by-step explanation:
A, B and C together can finish a piece of work in 4 days, A alone can do it in 12 days and B alone in 18 days. How many days will be taken by C to do it alone?
Answer:
9 days
Step-by-step explanation:
Let x = days taken by C to do it alone
In one day, A can do: 1/12 (part of the whole work)
In one day, B can do: 1/18 (part of the whole work)
In one day, C can do: 1/x (part of the whole work)
In one day,together A, B and C can do: 1/4 (part of the whole work)
we have this equation:
1/x + 1/12 + 1/18 = 1/4
--> 1/x = 1/4 - 1/12 - 1/18 = 1/9
--> x = 9 (days)
what is x times negative 2
Answer:
-2x
Step-by-step explanation:
Tony rounded each of the numbers 1,143 and 1,149 to the nearest hundred. Which choice correctly compares the rounded numbers?
The rounding off should be 1100.
What is rounding off?Rounding a number means the process of making a number simpler such that its value remains close to what it was. The result obtained after rounding off a number is less accurate, but easier to use. While rounding a number, we consider the place value of digits in a number.
Rounding Numbers to the Nearest HundredRounding numbers to the nearest hundred means we need to check the digit to the right of the hundreds place, that is, the tens place. For example, when 7456 is rounded to the nearest hundred, it becomes 7500.
Given numbers are 1143 and 1149.
According to the given numbers both the numbers rounded off nearest hundred.
The number 143 and 149 lies between 100 to 200.
So, the round off will be 100.
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You are working with a quadratic equation and construct the following table.
Answer:
a) Vertex is at (-3, -1)
b) y-intercept is at (0,8)
c) x intercept is at (-4,0) and (-2,0)
d) x=-3
Step-by-step explanation:
We know the vertex is the lowest or highest part of a parabola meaning all the points are reflected across. It is the only y value without a matching pair
E.x. -1 and 1, -3 and 3
The only point without a corresponding y value is (-3,-1) therefore the vertex is at (-3,-1)
The y-intercept is where the parabola meets the y axis or the x value is equal to 0, we just have to find when x = 0 to find the y intercept
y intercept: (0,8)
For the x intercept's it is just the reverse, you need to find where the parabola crosses the x axis or when y = 0
x intercept 1:(-4,0)
x intercept 2: (-2,0)
The axis of symmetry is also the x coordinate of the vertex which is 3 so
x = 3
A company has determined that 1% of its widgets are defective. If a customer received 2 widgets, what is the probability that they are both defective?
Assume independence because there are probably a lot of widgets. 1/100 x 1/100 = 1/10,000
Answer:
I'm just answering so the other person can have brainliest.
Sometimes I wonder if I'm the only idiot in the world.
Step-by-step explanation:
10,13,16,19
What is the 30th term?
Step-by-step explanation:
An=A1+(n-1)d
A30=10+(30-1)3
A30=10+(29)3
A30= 10+87
A30=97
it takes me a centuary to type
Find the inverse of the exponential function f(x)=10^x
Answer: [tex]y=\frac{ln(x)}{ln(10)}[/tex]
Step-by-step explanation:
To find the inverse of the function, you first replace the y with x and x with y. Then you solve for y.
[tex]y=10^x[/tex] [replace y with x and x with y]
[tex]x=10^y[/tex] [find natural log of both sides to get y alone]
[tex]ln(x)=ln(10)^y[/tex] [divide ln(10) on both sides]
[tex]\frac{ln(x)}{ln(10)} =y[/tex]
Now that we have found y after switching y with x and x with y, we know that the inverse is [tex]y=\frac{ln(x)}{ln(10)}[/tex].
What are the most appropriate measures of center and spread of the data set?
Answer:
B... According to the statement..
Step-by-step explanation:
The mean and median determines the numerical measures of center. The mean is used when showing the center as an average while the median shows the middle value of a data. The range and IQR determines the numerical measures of spread.
Hope it helped u if yes mark me BRAINLIEST!
Tysm!
Answer:
answers below vote brainliest
Step-by-step explanation:
f(x) = -x - 3 Find f(-4) f(-4) = -(-4) -3
Answer:
[tex] \boxed{ \bold{ \boxed{ \sf{f( - 4) = 1}}}}[/tex]
Step-by-step explanation:
Given, f ( x ) = - x - 3
Let's find the value of f ( - 4 )
[tex] \sf{f( - 4) = - ( - 4) - 3}[/tex]
We know that , [tex]( - ) \times ( - ) = ( + )[/tex]
⇒[tex] \sf{ 4 - 3}[/tex]
Subtract 3 from 4
⇒[tex] \sf{1}[/tex]
Hope I helped!
Best regards!!
Simplify the following expression as much as possible. 4^10/4^10 x 7^0=?
Answer:
1
Step-by-step explanation:
4^10/4^10 x 7^0 =
= 4^(10 - 10) * 1
= 4^0 * 1
= 1 * 1
= 1
Step-by-step explanation:
divide 4 to the tenth with 4 to the tenth first
the multiply it with 7 to the zero.
( 0 as an exponent is one)
ANSWER: 1
Write an equivalent expression by distributing the "-−" sign outside the parentheses: 2h-(-2.3k-4) 2h−(−2.3k−4)
Answer:
[tex]2\cdot h +(2.3\cdot k + 4)[/tex]
Step-by-step explanation:
This expression is solved algebraically:
1) [tex]2\cdot h -(-2.3\cdot k - 4)[/tex] Given
2) [tex]2\cdot h +(-1)\cdot [(-1)\cdot (2.3)\cdot k + (-1)\cdot (4)][/tex] [tex](-1)\cdot a = -a[/tex]
3) [tex]2\cdot h + [(-1)\cdot (-1)]\cdot (2.3\cdot k+4)[/tex] Distributive property.
4) [tex]2\cdot h +(2.3\cdot k + 4)[/tex] [tex](-a)\cdot (-b) = a\cdot b[/tex]/Modulative property/Result
PLEASE HELP, I DON'T UNDERSTAND THIS! :(
Divide.
Answer:
Step-by-step explanation:
Hello, please consider the following.
We need to factorise as much as possible and eliminate the same term from numerator and denominator
[tex]\dfrac{\dfrac{x^2-4x-5}{x-2}}{\dfrac{2x-10}{x^2-4}}\\\\\\=\dfrac{(x^2-4x-5)}{(x-2)}\dfrac{(x-2)(x+2)}{(2x-10)}\\\\\\=\dfrac{(x-5)(x+1)(x+2)}{(2x-10)}\\\\\\\large \boxed{=\dfrac{(x+1)(x+2)}{2}}[/tex]
Thank you
The student council at Mountain View High School plans to conduct a survey to determine the student body's opinion of the new lunch menu. Which of the following sampling methods would be most likely to result in nonresponse bias? O A. Handing out a questionnaire to seniors only 0 B. Obtaining an alphabetical list of all students and surveying every 10th student O C. Setting up a booth in the lunchroom where students can give their thoughts D. Handing out a questionnaire to students leaving campus at lunchtime
Answer:
C. Setting up a booth in the lunchroom where students can give their thoughts
Step-by-step explanation:
Sampling is the process by which some part of the entire population is selected and researched. This selected population helps in estimating the researches for the entire population. There are various methods used for the process of sampling.
For the given case, the sampling method to be adopted is to view each students' opinion on the issue. For this process, setting up a booth can prove helpful as the opinion of every student can be viewed with no biases. Also, every student is free to provide their opinion with any fear.
5. For a given function ƒ(x) = x^5, a translation in the positive y-direction of two units is represented by which of the following?
options:
A) ƒ(x + 2) = (x + 2)^5
B) ƒ(x + 2) = ( x – 2)^5
C) ƒ(x) – 2 = x^5 – 2
D) ƒ(x) + 2 = x^5 + 2
Answer:
[tex]\huge \boxed{\mathrm{Option \ D}}[/tex]
Step-by-step explanation:
[tex]f(x)+k[/tex]
Is the formula for upward vertical translation.
[tex]f(x)-k[/tex]
Is the formula for downward vertical translation.
Where [tex]k[/tex] is the units being translated.
Translation in the positive y-direction is upward vertical translation.
[tex]k=2[/tex]
The translation is by 2 units.
The function becomes,
[tex]f(x)+2[/tex]
Answer:
Step-by-step explanation:
D) ƒ(x) + 2 = x^5 + 2
The function h(x) = 5x2 – x – 6 is the result of combining two functions using multiplication. Which functions could have been combined? f(x) = 5x – 6 and g(x) = x + 1 f(x) = 5x – 2 and g(x) = x + 3 f(x) = 5x3 + 14x2 – 9x – 18 and g(x) = x + 3 f(x) = 25x3 – 15x2 – 28x + 12 and g(x) = 5x – 2
Answer:
f(x) =5x -6 and g(x) = x + 1
Step-by-step explanation:
5x^2 - x - 6
finding the product of AC where A is 5 and C is -6 equals -30
factors of -30 whose sum eqjals -1 are +5 and -6
substitute into the expression
5x^2 + 5x -6x - 6
group the expression
(5x^2 + 5x) (-6x - 6)
take out the common factors in each bracket
5x(x + 1) -6 (x + 1)
(5x - 6) (x + 1)
Answer:a
Step-by-step explanation:
Round 3.99408317124 to the nearest tenth.
Answer:
4.0
Step-by-step explanation:
when rounded the nine adds to the whole number making it 4
Rounding 3.99408317124 to nearest tenth becomes 4.
Why we round numbers ?We round numbers as it is easy to remember and also rounding a number to a certain place retains approximate value to that pace.
According to the given question we have to round 3.99408317124 to nearest tenth.
We know to round a number to a certain place we have to check the
digit that is one place lesser.
Now we have to round 3.99408317124 to tens place so we'll look at the digit at ones' place which is 9 and is greater than 5 so we'll add one to it this will make it 0 and 1 will carry over.
Thus the number becomes 4.0.
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Which of the following is equivalent to √8?
A.2
B.√2
C.2 √2
D.2 + √2
Answer:
[tex]C.2\sqrt{2}[/tex]
Step-by-step explanation:
Step 1: Know you can split 8 into factors of 4 and 2
[tex]\sqrt{(4)(2)}[/tex]
Step 2: Recognize 4 is a square number
[tex]\sqrt{(2^{2})(2) }[/tex]
Step 3: Because [tex]2^{2}[/tex] is squared you can move it out
[tex]2\sqrt{2}[/tex]
Therefore the answer is C
Answer this question please. Thanks
Answer:
a). $8.30
b).$18.70
c). $10.40
d). 1.04
Step-by-step explanation:
Average price of the ticket of a basketball game can be represented by the function,
p(x) = 0.03x² + 0.56x + 6.35
Where 'x' is the number of years or duration.
a). P(3) = 0.03(3)² + 0.56(3) + 6.35
= 0.27 + 1.68 + 6.35
= $8.30
b). P(13) = 0.03(13)² + 0.56(13) + 6.35
= 5.07 + 7.28 + 6.35
= $18.70
c). P(13) - P(3) = 18.70 - 8.30
= $10.40
d). [tex]\frac{P(13)-P(3)}{13-3}[/tex] = [tex]\frac{10.40}{10}[/tex]
= 1.04
Average increase in the price of each ticket from 3 tickets to 13 tickets is $1.04.
f + 3 pi = 7 pi what is f?
Answer:
f = 4 pi
Step-by-step explanation:
f + 3 pi = 7 pi
f = 7 pi - 3 pi
f = 4 pi
Answer:
f = 4 pi
Step-by-step explanation:
f + 3 pi = 7 pi
Subtract 3 pi from each side
f + 3 pi -3pi = 7 pi -3 pi
f = 4 pi
How many ways can six of the letters of the word ALGORITHM be selected and written in a row if the first three letters must be THM
Answer: 120 ways
Step-by-step explanation: The word ALGORITHM has 9 letters. We want them to be written in a row, which means, order of the letters are important, so, use Permutation.
Permutation is calculated by:
P(n,r) = [tex]\frac{n!}{(n-r)!}[/tex]
We want to use 6 of the 9 letters with first 3 being THM. So, n = 6 and r = 3:
Permutation will be:
P(6,3) = [tex]\frac{6!}{(6-3)!}[/tex]
P(6,3) = [tex]\frac{6.5.4.3!}{3!}[/tex]
P(6,3) = 120
There are 120 ways six letters of ALGORITHM be written in a row with first three letters THM.
Jessie leaves home at 7:30 am and rides his bike to school at a speed of 9 miles per hour. If his school is 3 miles from his house, what time will he get to school?
Step-by-step explanation:
d =r*t
d=3 mi
r =9 mi/hr
3=9t
t=1/3 hr
Convert hour to min
t=20 min added to 7:30
The time will he get to school is 7:50 AM
In which step was the addition property of equality applied? A. step 2 B. step 3 C. step 4 D. The addition property of equality was not applied to solve this equation.
Answer:
Step 2
Step-by-step explanation:
The addition property of equality was applied in step 2. The '- 9' value had to be 'undone', so '9' was added to each side of the equation, which would be an application of the addition property of equality, to yield step 3.
Hope this helps.
The correct option is A. Step 2.
Given the image in question as we saw in step 2, 9 has been added to both sides of the equation,right hand side and left hand side.
Hence in step 2 the addition property of equality applied.
So, the correct option is A. step 2.
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