Can anyone explain why this two is being added before the integral? Does it have something to do with integral symmetry?
Answers
Yes it's exactly due to symmetry. Specifically, symmetry about the x axis.
Simply writing \(\displaystyle \int_{-8}^{-4}\left[\sqrt{x+8} \ \right]dx\), without the 2 out front, will only get you the area of the portion shown in red (see diagram below).
The blue region has equal area of the red region due to symmetry.
So,
\(\displaystyle 2\int_{-8}^{-4}\left[\sqrt{x+8} \ \right]dx\)
represents the red and blue regions combined.
The portion in green is \(\displaystyle \int_{-4}^{8}\left[\sqrt{x+8} - \frac{x}{2} \ \right]dx\) which is the integral of the difference of the upper and lower curves over the interval \(-4 \le x \le 8\)
---------------
In all honesty, it's probably easier to integrate with respect to y since the given functions are in terms of y initially. Also, there isn't a junction point in which the curves swap places in terms of which one is larger. However, it doesn't hurt to have practice in integrating with respect to x.
If you're curious about what the y integral looks like, then it would be
\(\displaystyle \int_{-2}^{4} \bigg[ (2y) - (y^2-8)\bigg] dy\)
You can use a tool like WolframAlpha to check that both integral expressions result in 36 to help confirm that they represent the same overall area (just in different ways of course).
Answer:
Yes - see attached sketch and explanation
Step-by-step explanation:
As you have a sideways parabola, you would need to split the integral into 3 parts.
Find the x coordinate where the parabola first intersects the line: x = - 4.
Now you should be able to see that the parabola between where it intersects the x-axis (x = -8) and the line x = -4 can be split into 2 equal parts (above and below the x-axis). I have shaded these parts as blue and red on my attached sketch.
As the areas above and below the x-axis and the parabola are equal, this can be written as 2 x the area ABOVE the x-axis (the positive integral). So yes, this is why the 2 has been added before the integral, as without the 2 the integral is finding the area ABOVE the x-axis and the parabola curve ONLY.
Then you would find the area between the parabola and the line between x = -4 and x = 8 (shaded grey on my sketch), and as it's only the positive part of the parabola, a normal "integral between 2 curves" can be applied, i.e.
\(\int\limits^a_b {(top-bottom)} \, dx \)
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qhubuag qjjbbaqbjbqhuqguqbqbqyqbyqbyqb math help
Answers
So, we know that JI and JX are perpendicular (first statement). That just means that they mean at an angle of 90*.
That means CIJ has an angle of 150*-90*=60*. Therefore, as IG bisects CIJ , both CIG and GIJ have angle 60*/2=30*.
Half a circle has 180*. GIR is just short of CIG to be a half circle.
Therefore
180*-mCIG=mGIR
We know mCIG was 30*, so
150*=mGIR
The graph shows the location of point P and point R. Point R is on the y-axis and has the same y-coordinate as point P. Point Q is graphed at (n,-2). The distance from point P to point Q is equal to the distance from point P to point R. What is the from point P to point Q? What is the value of N? Explain how you determined the distance from point P to point Q, and the value of N.
Answers
The value of n such that the conditions are met is:
\(n = \frac{2x \pm \sqrt{(-2x) - 4*1*(y + 2)^2} }{2}\)
Where x and y are the coordinates of point P.
How to get the given points?We know that:
Q = (n, -2)R = (0, y) (because R is on the y-axis).P = (x, y)Now, the distance of P to Q is equal than the distance of P to R, then we have that:
\(\sqrt{(x - 0)^2 + (y - y)^2} = \sqrt{(x - n)^2 + (y + 2)^2} \\\\x^2 = (x - n)^2 + (y + 2)^2\)
So we need to solve this for n.
\(x^2 = x^2 - 2xn + n^2 + y^2 + 4y + 4\\\\0 = -2xn +n^2 + (y + 2)^2\\\\\\0 = n^2 - (2x)*n + (y + 2)^2\)
Notice that this is a quadratic of n, the two solutions of n will be:
\(n = \frac{2x \pm \sqrt{(-2x) - 4*1*(y + 2)^2} }{2}\)
Such that, as you can see, the value of n depends on x and y.
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The Karns Recreation Hall needs to build a ramp. The height of the ramp must be 2 feet. The ramp will start 6 feet from the door. To the nearest tenth of a foot, how long will the ramp be?
Answers
Using the Pythagorean's theorem we can see that the length of the ramp is 6.32ft
How long the ramp will be?We know that the ramp is a right triangle whose legs are 2 ft and 6 ft. Remember that by Pythagorean's theorem we know that the square of the hypotenuse is equal to the sum of the squares of the legs.
Then the length of the ramp L will be:
L² = (2ft)² + (6ft)²
L = √((2ft)² + (6ft)²)
L = 6.32ft
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PLEASE HELP ASAP!!
Shelby bought a circular spin wheel to make a game for the end-of-year carnival. She knows
that a circle has 360 degrees in all. She wants to paint 24 equal-sized wedges on the wheel.
Let w represent the angle measure of each wedge. Which equation models the problem?
W/24= 360 or
24w = 360
Solve this equation to find the angle measure of each wedge.
Answers
As a result, every single wedge on the cylindrical spin wheel needs to be 24 degrees in inclination.
What equation of number would that be?The meaning of an equation in algebra is a mathematical assertion that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two expressions 3x + 5 and 14, which are split by the 'equal' symbol.
The equation that models the problem is:
w/24 = 360/360
or simply:
w/24 = 1
To solve for w, we can multiply both sides of the equation by 24:
w/24 × 24 = 1 × 24
w = 24
Therefore, each wedge on the circular spin wheel should have an angle measure of 24 degrees.
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I Need Help Please This Will Only Take 1min…
Answers
The (C) solution of the equation is the value of the variable that causes an equation to be a true statement.
What are equations?A mathematical statement that has two expressions with equal values separated by the symbol "equal to" is called an equation. Consider the formula 3x + 5 = 15. Different types of equations exist, including linear, quadratic, cubic, and others.So, the Root or solution of the equation is the value of the variable that turns an equation into a true statement.
An equation that compares two mathematical expressions is a sentence or mathematical statement.The value of the variable that makes the equation true or satisfies it is referred to as the equation's root or solution.For instance,
"x + 2 = 0"The equation's solution in this case is x = - 2.
x² - 1 = 0Here, the equation's roots or solutions are x = 1 and -1.
Therefore, the (C) solution of the equation is the value of the variable that causes an equation to be a true statement.
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The complete question is given below:
The value of the variable which makes an equation a true statement is called the ________ of the equation.
A. Coefficient
B. Equation
C. Solution
D. Algebraic expression
E. Constant
F. Variable
pls help me i’ll give ty brainlist
Answers
\(\sqrt{25*7x^{4}*x^{1} } \\=\sqrt{25*x^{4}*7 *x^{1} } \\=\sqrt{25x^{4} } *\sqrt{7x}\\=5x^{2} \sqrt{7x}\)
Option B is the answer.
Answer: C
i hope you can see my handwriting
Order the side lengths HI, IK, HJ, IJ, and K J from least to greatest.
Answers
Answer:
I think it's IH, IJ, JK, HJ, IK.
What is the linear function equation that best fits the data set?
ŷ = x – 2
ŷ = 4x – 11
ŷ = 2x – 2
ŷ = 4x – 2
Answers
Answer:
Option 3
Step-by-step explanation:
Given data set is,
{(1.5, 1), (2.5, 3), (3, 3), (3, 5), (3.5, 5), (4.5, 7), (5, 7), (5, 9), (5.5, 9), (6.5, 11), (7.5, 13)}
By using linear regression calculator,
Equation of the linear function will be,
y = 2x - 2
Therefore, Option 3 will be the correct option.
Marilyn is buying a birthday present for her son from his favorite store. She has a $25 store credit that will help cover the cost of his gift. Marilyn wants to spend less than $31 after applying the store credit. If p represents the cost of the present, then this situation can be modeled by the inequality,
p - 25 < 31
Which value could be a possible cost for the present and a solution to Marilyn’s inequality?
Answers
Therefore , the solution of the given problem of inequality comes out to be less than 56, such as $30 or $45.
Inequality: what is it?Due to the lack of an indication for this distinction in algebra, it can be represented by a combination or set two numbers. Equilibrium is typically followed by equity. The ongoing disparity in standards is the source of inequality. Disparity is not the same as equality. Despite knowing that the elements are frequently not related nor close to one another, as was my least favourite symbol. (). All deviations, no matter how slight, have an impact on value.
Here,
We must isolate the variable p on one side of the inequality sign in order to solve the inequality:
=> p - 25 < 31
To both sides, add 25:
=> p - 25 + 25 < 31 + 25
Simplify:
=> p < 56
Since the cost would be less than $31 after applying the shop credit, any value of p that is less than 56 would constitute a solution to Marilyn's inequality.
When the store credit is applied, Marilyn would pay $25 rather than $31 for a $50 gift, for instance.
Therefore, $50 is a potential expense for the present as well as a remedy for the inequality.
There are other other solutions that are less than 56, such as $30 or $45.
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Identify the dividend.
35 ÷ 5 = 7
A. 35
B. 5
C. 7
Answers
Answer:
35
Step-by-step explanation:
35 is the number being divided
35/5=7
35
Amadi has jar with one dollar and two dollar coins in it.
There are coins in total.
A jar of one and two dollar coins.
The ratio of one dollar coins to two dollar coins is .
How many coins are two dollar coins
Answers
Without knowing the total number of coins or the ratio between one dollar and two dollar coins in the jar, we cannot accurately determine the number of two dollar coins.
Amadi has a jar with both one dollar and two dollar coins. The question is asking how many of the coins in the jar are two dollar coins.
To determine the number of two dollar coins, we need more information. The question does not provide any details about the total number of coins or the ratio between one dollar and two dollar coins in the jar. Without this information, it is not possible to give an accurate answer.
If we assume that the jar contains a total of 100 coins, we can use algebra to solve for the number of two dollar coins.
Let's say the number of two dollar coins is x. Then, the number of one dollar coins would be 100 - x.
Since the value of the two dollar coins is double the value of the one dollar coins, we can set up the equation 2x + 1(100 - x) = total value of coins.
Simplifying the equation, we get 2x + 100 - x = total value of coins.
Combining like terms, we have x + 100 = total value of coins.
Since we don't know the total value of coins, we cannot solve for x. Therefore, we cannot determine the number of two dollar coins without additional information.
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In the equation y = mx + b, the "b" is the y-intercept
True
False
Answers
Answer:
False
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A novelist can write 2 1/4 pages in 3/5 of an hour what is her sped in her pages per hour x
Answers
Answer:
The novelist can write 4/15ths of a page in an hour.
Step-by-step explanation:
Just multiply 3/5 by the reciprocal of 9/4, which is 4/9. Multiply 3/5 and 4/9 to get 12/45 or 4/15. In conclusion, the writer can write 4/15th of a page in an hour.
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Question 1 (Multiple Choice Worth 2 points)
(Effects of Changes in Data MC)
The heights of 11 plants, in inches, are listed.
14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 22
If another plant with a height of 14 inches is added to the data, how would the range be impacted?
Answers
The range of the data is the difference between the highest and lowest values, which is 8 inches. Adding another plant with a height of 14 inches would not change the range, so the range would remain 8 inches.
What is data?Data is a collection of facts, figures, or information that can be processed and used to draw conclusions or make decisions. It can be stored in a variety of formats, including numerical, text, images, audio, and video. Data can be collected from a variety of sources, such as surveys, observations, experiments, or transactions. It can be used to inform decisions in areas such as business, health care, education, and research. Data can be analyzed to gain insight, develop strategies, identify trends, and make predictions. Data is a valuable resource and is increasingly important in today's digital age.
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write an equation perpendicular to y = 3/4x that passes through the point(3,-3)
Answers
Step-by-step explanation:
the slope perpendicular to y= ¾x is
-1 ÷ ¾ = -4/3
so the equation :
y-(-3)= -4/3(x-3)
y+3= -4/3x +4
y = -4/3x +4-3
y = -4/3x + 1
Can someone help me with this math problem?
see attached photo
Answers
By applying definitions of trigonometric reasons, Pythagorean theorem and a trigonometric expression, the value of the sin 2θ is equal to - (5/18) · √11.
How to determine the value of trigonometric reasons
In this question we must take into accounts the definitions of the trigonometric reasons sine and secant and the Pythagorean theorem as well.
If sec θ < 0 and csc θ > 0, then sin θ > 0 and cos θ < 0, the angle is in the second quadrant (0.5π < θ < π, x < 0, y > 0) and we have the following expression:
sec θ = r/x
\(\frac{r}{x}= -\frac{6\sqrt{11}}{11}\)
Then x = - 11, r = 6√11 and the value of y is:
\(y = \sqrt{r^{2}-x^{2}}\)
y = 5√11
By trigonometric expressions we have this formula for sin 2θ:
sin 2θ = 2 · sin θ · cos θ
sin 2θ = 2 · (y/r) · (x/r) = (2 · x · y)/r²
sin 2θ = [2 · (- 11) · (5√11)]/396
sin 2θ = - (5/18) · √11
By applying definitions of trigonometric reasons, Pythagorean theorem and a trigonometric expression, the value of the sin 2θ is equal to - (5/18) · √11.
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Commitment to goals can be promoted in which of the following ways?
Answers
Commitment to goals can be promoted in through:
by imagining how you will feel when you achieve themby identifying who or what might keep you from achieving themWhat is goal commitment?Goal commitment is the degree of determination a person uses to achieve an accepted goal, and it is determined by two major factors: importance and self-efficacy. The reasons a person has for achieving a goal, including expectations of certain outcomes, are significant. Commitment helps you stick to your goals in both good and bad times, when obstacles arise. Commitment is influenced by two factors: importance and
Goal commitment is generally defined as an individual's determination to devote time and effort to achieving a goal.
Goal commitment is the degree of determination a person uses to achieve an accepted goal, and it is determined by two major factors: importance and self-efficacy. The reasons a person has for achieving a goal, including expectations of certain.
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HELP! QUICK I NEED HELP ON THIS
Answers
Answer:
2) definition of midpoint
4) RS=RS, reflexive property
5) SSS theorem
6) CPCTC
A rectangular painting has
sides that measure 7
inches and 6 inches.
What is the area of this
rectangular painting?
Answers
Answer:
42 square inches.
Step-by-step explanation:
The formula for area is Length x Width
7 x 6 = 42
Then add the units to be specific, and you get:
42 square inches
Which of the following is described below?
This object is infinitely small. It is
comprised of a single location.
B. Line
A. Point
D. Plane
C. Ray
Answers
Answer:
A. Point
Step-by-step explanation:
A point is infinitely small and comprised of a single location.
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The object that is infinitely small among the option is a point. Option A is correct.
What is standard measurement?All over the world, most things are in use as of common quantity, so people all over the world agree that they have to follow a common system of measurement and that the common system of measurement is called standard measurement.
Here,
The object that is infinitely small among the option is a point. Because there is no instrument that can measure the point. Since a point is nothing but a demonized dot. While line, ray, or plane are infinitely long and their measure can be possible.
Thus, the object that is infinitely small among the option is a point. Option A is correct.
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Only 3 more questions to go
Answers
Determine whether the systems have one solution, no solution, or infinitely many solutions 5-y=2-y+3
Answers
Answer:First system: no solution
Second system: one solution
Third system: one solution
Fourth system: infinite solutions
Fifth system: no solution
Step-by-step explanation:
suppose you want to enlarge a 4-inch by 6-inch photograph as much as possible so that it fits on a bulletin board that is 24 inches high. what is the widest the photograph can be?
Explain how you could find the measurements of an enlargement if you know the ratio of the measurements of the original to the measurements of the enlargement.
answer fast please
Answers
The widest the photograph can be when enlarged to fit on a 24-inch high bulletin board is 36 inches.
How to determine the maximum width of the photograph when enlarged
First we need to find the scale factor of the enlargement.
If we let x be the width of the enlarged photograph, then we can set up the proportion:
6/4 = x/24
This is saying that the ratio of the width to height of the original photograph (6/4) is equal to the ratio of the width to height of the enlarged photograph (x/24).
To solve for x, we can cross-multiply:
6 * 24 = 4 * x
144 = x * 4
x = 36
Therefore, the widest the photograph can be when enlarged to fit on a 24-inch high bulletin board is 36 inches.
If we know the ratio of the measurements of the original to the measurements of the enlargement, we can use a similar approach to find the measurements of the enlargement. For example, if the ratio is 1:2 (meaning the enlargement is twice as big as the original), we can multiply each dimension of the original by 2 to find the corresponding dimension of the enlargement.
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Evaluate −a2+c2 when c=−4.
Answers
Answer:
\(a = 4, -4\)
Step-by-step explanation:
Step 1: Plug in -4 for c
\(-a^{2} + c^{2}\)
\(-a^{2} + (-4)^{2}\)
\(-a^{2} + 16\)
Step 2: Solve for a
\(-a^{2}+16-16=0-16\)
\(-a^{2}/-1 = -16/-1\)
\(a^{2} = 16\)
\(\sqrt{a^{2}} = \sqrt{16}\)
\(a = 4, -4\)
Answer: \(a = 4, -4\)
Suppose that in an election for governor of Oregon there are five candidates of whom two are women. A statistics student reasons as follows: the probability that a woman will win the election is equal to 2/5. What is wrong with this reasoning?
Answers
Answer:
If in an Oregon state gubernatorial election, 2 of the 5 candidates are women, and a statistics student establishes that there is a 2/5 chance that the person elected for governor is a woman, this reasoning will be wrong.
This will be the case because not all candidates have the same amount of votes, that is, not all start with the same conditions: thus, there are more popular candidates than others, who will have greater probabilities of being elected as governors, regardless of their sexual identity.
What is the solution set for 7x − 3 =18, given the replacement set {0, 1, 2, 3, 4}?
x = 0
x = 1
x = 2
x = 3
x = 4
Answers
Answer:
x = 3
Step-by-step explanation:
Let's solve your equation step-by-step.
7x−3=18
Step 1: Add 3 to both sides.
7x−3+3=18+3
7x=21
Step 2: Divide both sides by 7.
7x/7 = 21/7
x=3
Write the slope of tangent and normal to the curve y=f(X) at (x1,y1).
Answers
The slope of the tangent line is:
a = (df(x1)/dx)
The slope of the normal one is:
a' = -1/(df(x1)/dx)
How to find the slope of the tangent line?For any function y = f(x), we define the slope of the tangent line at the point x as the derivative evaluated in x.
So if we want to find the slope of the tangent line at the point (x1, y1), we just need to differentiate and evaluate in x1, then the slope will be given by:
df(x1)/dx
And the normal slope is the slope of the perpendicular line, two slopes are perpendicular if the product is -1, then:
a*(df(x1)/dx) = -1
a = -1/(df(x1)/dx)
Where df/dx reffers to the derivative of f(x).
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mations to Determine Similarity
What steps would you take to determine if these
figures are similar? Check all that apply.
Use a scale factor of 2.
Multiply the vertices of polygon ABCD by
O Translate the intermediate image 4 units down.
Perform two different dilations.
Reflect the intermediate image.
Answers
The steps which you would take to determine if these figures are similar include the following:
2. Multiply the vertices of polygon ABCD by 1/2.
5. Reflect the intermediate image.
What is a dilation?In Mathematics and Geometry, a dilation is a type of transformation which typically transforms the dimension (size) or side lengths of a geometric object, without affecting its shape.
Therefore, the dimension or side lengths of the dilated geometric object would be stretched or compressed (shrunk) depending on the scale factor that is applied.
By critically observing the graph representing polygon ABCD, we can logically deduce that a sequence of transformations that would polygon ABCD (pre-image) onto polygon A"B"C"D" (image) is a dilation by a scale factor of 1/2 and a reflection of intermediate image.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Wally completed 7 out of 10 baskets. What percent is this?
Answers
Answer:70%
Step-by-step explanation:you have 7/10.SO u times the whole fraction by 10( 7 times 10 =70 and 10 times 10 =100) which means 70/100 so the percent is 70.
Hope it helps
write y+4=-2(x-1) in slope intercept form
Answers
Answer:
y=2x-6
Step-by-step explanation:
y+4=-2(x-1)
Since the slope intercept form is in the form of:
y=mx+c
Making above equation in this form.
y+4=-2(x-1)
opening bracket
y+4=2x-2
subtracting both side by 4.
y+4-4=2x-2-4
y=2x-6
This equation is the slope intercept form.