1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alina [70]
2 years ago
9

Can anyone explain why this two is being added before the integral? Does it have something to do with integral symmetry?

Mathematics
2 answers:
PIT_PIT [208]2 years ago
5 0

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).

statuscvo [17]2 years ago
3 0

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

You might be interested in
Help me please!!!!!!!
Cerrena [4.2K]

Answer:

I have no idea.

Step-by-step explanation:

I failed geometry.

3 0
3 years ago
Use the Venn diagram about students’ pets to solve. How many students do not have a cat or a dog?
Gennadij [26K]
There is no Venn diagram so how do we solve it. There is nothing to help us with this answer so right now there is no definite amount.
6 0
3 years ago
What is the area of the figure?<br> Enter your answer in the box.
nordsb [41]

Answer:

The area of the figure = 112 in²

Step-by-step explanation:

The figure consist of:

1) A square of side length of 10 in.

2) A right triangle with a base of (10 - 6 = 4 in) and a height of (16 - 10 = 6 in)

So,

The area of the square = (side length)² = 10² = 100 in²

The area of the triangle = 0.5 * height * base = 0.5 * 6 * 4 = 12 in²

So,

The total area = 100 + 12 = 112 in²

<u>The area of the figure = 112 in²</u>

6 0
3 years ago
Recommendations
Ipatiy [6.2K]

Answer:

4

Step-by-step explanation:

you're just going to plug in 15 where f is so its asking 15-11 which would be four

4 0
3 years ago
As an estimation we are told £3 is €4. <br> Convert £22.50 to euros.
Talja [164]

Answer:

€16.875 is the answer

Step-by-step explanation:

£22.50/3*100

=16.875

5 0
2 years ago
Other questions:
  • Find the area of the figure. Round to the nearest tenth.
    6·1 answer
  • If f(x)=-5^x-4 and g(x)=-3x-2 find (f-g)(x)
    14·2 answers
  • The following price reductions are available
    11·1 answer
  • Find the value of t. Write your answer as a decimal number. 88.65 t 88.75 on a number line?
    6·1 answer
  • Find each percent of change. 105 is decreased to 32
    6·1 answer
  • Natasha can type 30 words a minute. <br> How many words can be typed in 7.5 minutes?
    5·2 answers
  • Twice h increase by 5
    6·1 answer
  • Zahir made this wooden perch for his pet bird. What is the volume of the bird perch? Round to the nearest tenth if necessary.
    10·1 answer
  • Please help!!! I will vote BRAINLIEST<br> What is the value of k?
    15·1 answer
  • 4 + 1/3. and 4/5 + 1/3
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!