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
atroni [7]
3 years ago
15

Hi guys, Can anyone help me with this tripple integral? Thank you:)

Mathematics
1 answer:
OleMash [197]3 years ago
6 0

I don't usually do calculus on Brainly and I'm pretty rusty but this looked interesting.

We have to turn K into the limits of integration on our integrals.

Clearly 0 is the lower limit for all three of x, y and z.

Now we have to incorporate

x+y+z ≤ 1

Let's do the outer integral over x.  It can go the full range from 0 to 1 without violating the constraint.  So the upper limit on the outer integral is 1.

Next integral is over y.  y ≤ 1-x-z.   We haven't worried about z yet; we have to conservatively consider it zero here for the full range of y.  So the upper limit on the middle integration is 1-x, the maximum possible value of y given x.

Similarly the inner integral goes from z=0 to z=1-x-y

We've transformed our integral into the more tractable

\displaystyle \int_0^1 \int_0^{1-x} \int _0^{1-x-y} (x^2-z^2)dz \; dy \; dx

For the inner integral we get to treat x like a constant.

\displaystyle \int _0^{1-x-y} (x^2-z^2)dz = (x^2z - z^3/3)\bigg|_{z=0}^{z= 1-x-y}=x^2(1-x-y) - (1-x-y)^3/3

Let's expand that as a polynomial in y for the next integration,

= y^3/3 +(x-1) y^2 + (2x+1)y -(2x^3+1)/3

The middle integration is

\displaystyle \int_0^{1-x} ( y^3/3 +(x-1) y^2 + (2x+1)y -(2x^3+1)/3)dy

= y^4/12 + (x-1)y^3/3+ (2x+1)y^2/2- (2x^3+1)y/3 \bigg|_{y=0}^{y=1-x}

= (1-x)^4/12 + (x-1)(1-x)^3/3+ (2x+1)(1-x)^2/2- (2x^3+1)(1-x)/3

Expanding, that's

=\frac{1}{12}(5 x^4 + 16 x^3 - 36 x^2 + 16 x - 1)

so our outer integral is

\displaystyle \int_0^1 \frac{1}{12}(5 x^4 + 16 x^3 - 36 x^2 + 16 x - 1) dx

That one's easy enough that we can skip some steps; we'll integrate and plug in x=1 at the same time for our answer (the x=0 part doesn't contribute).

= (5/5 + 16/4 - 36/3 + 16/2 - 1)/12

=0

That's a surprise. You might want to check it.

Answer: 0

You might be interested in
How do the 8s in 6.58 and 47.816 compare?
il63 [147K]

Answer:

C

Step-by-step explanation:

0.08 multiplied by 10 is 0.8.

4 0
3 years ago
A necklace usually sells for 18, but is onsale for 15% off. What is the sale price of the necklace?
I am Lyosha [343]

Answer:

15.3

Step-by-step explanation:

18-(18×(15%))= 15.3

6 0
2 years ago
Read 2 more answers
What is 6 (5x + 2y) -3 (2x + 3y) ?
melisa1 [442]

Answer:

=24x+3y

Step-by-step explanation:

6(5x+2y)−3(2x+3y)

Distribute:

=(6)(5x)+(6)(2y)+(−3)(2x)+(−3)(3y)

=30x+12y+−6x+−9y

Combine Like Terms:

=30x+12y+−6x+−9y

=(30x+−6x)+(12y+−9y)

=24x+3y

3 0
4 years ago
Read 2 more answers
What is equivalent to 6/8?
patriot [66]
3/4
75%
0.75
75/100
theres a lot
4 0
3 years ago
Kayla brought 80 packs of cat food to feed the cats outside.
Akimi4 [234]

Answer:

It's 24 packs of cat food

Step-by-step explanation:

You have to think of it as 30% of 80.

30      ?

----- = ----

100    80

Now you have to find out what is in the blank by doing 80 times 30 to find what that equals. That equals 2,400. Now you have to find what times a 100 gets you 2,400. You divide a 100 by 2,400 to get 24. There's your answer

5 0
3 years ago
Read 2 more answers
Other questions:
  • Ab¯¯¯¯¯ is congruent to cd¯¯¯¯¯. point p is the midpoint of ab¯¯¯¯¯, and point q is the midpoint of cd¯¯¯¯¯. which relationship
    15·1 answer
  • Can anyone help me with this? I've been stuck on this question for some time-
    15·1 answer
  • Electric charge is distributed over the rectangle 2 ≤ x ≤ 4, 0 ≤ y ≤ 2 so that the charge density at (x, y) is σ(x, y) = 2xy + y
    14·2 answers
  • Compare 7/8 with 14/16 using (<,>,=)
    6·2 answers
  • -1/3 multiplied by 1/4
    8·1 answer
  • Use the distributive property to find the product. -x^2(x+5)
    5·2 answers
  • 8x2 + (- 14x2) – 16x2
    10·2 answers
  • How is 4.209 written using words? plz plz plz
    8·2 answers
  • What dose 2 + 2 = ?????????
    14·1 answer
  • In his garden Tim plants the seed 3 1/2 inches below ground. After one month the tomato plant has grown a total of 10 1/4. How m
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!