I'm given the region in the first octant bounded by z=sqrt{x^2+y^2}, z=sqrt{1-x^2-y^2}, y=x and y=sqrt3 x and I need to evaluate int int int_V dV

Violet Woodward 2022-07-17 Answered
Volume Integration of Bounded Region
I'm trying to integrate this volume in spherical and cylindrical coordinates, but having difficulty finding my bounds of integration;
I'm given the region in the first octant bounded by z = x 2 + y 2 , z = 1 x 2 y 2 , y = x and y = 3 x and I need to evaluate V d V.
When proceeding to integrate with spherical and cylindrical coordinates I am not getting the right bounds such that both methods equate to the same volume? I am definitely missing something. Any and all advice would be much appreciated!
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

yermarvg
Answered 2022-07-18 Author has 19 answers
Step 1
This region seems better defined using spherical coordinates than cylindrical. We are given that the region is between two vertical planes y = x and y = 3 x, and it is between the sphere x 2 + y 2 + z 2 = 1 and the upper half of the cone x 2 + y 2 = z 2 .
Step 2
From this, we can set the bounds to be:
π 3 θ π 4
from the region of angles between the two lines (arctan of root(3) is pi/3) 0 ϕ π 4 from the intersection of the cone and sphere 0 r 1 from the radius of sphere

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-21
Finding value of t such that volume contained inside the planes is minimum
The question is to find out the value of t such that volume contained inside the planes 1 t 2 x + t z = 2 1 t 2 ,
z = 0 ,
x = 2 + t 4 t 2 5 t + 2 12 ( 1 t 2 ) 1 / 4  and 
| y | = 2 is maximum.
I tried to figure out the line of the intersection of the planes but it made the problem more complicated.Is there any way by which the equations can be decomposed to make it easier to handle?Any help shall be highly appreciated.
asked 2022-09-13
A cylinder and a cone have a common base and height. Calculate the volume of the cylinder if the volume of the cone = 10.
asked 2022-08-11
Finding volume of solid revolution about x-axis
I'm asked to find the volume of a container of height 13.5 The container is made by rotating y = 1.5 x 2 where 0 x 3 about the axis x = 0.5 (the bottom is flat).
I tried finding the right volume a number of ways, but I believe my most solid attempt so far is this:
I invert the function to get y = 2 3 x .
From this I tried to calculate π 0 3 2 3 x d x which gave me 3 π which I thought was correct, but it wasnt. Plotting this, I realized the height becomes 3 when I integrate the inverted function from 0 to 3, so I thought I should maybe integrate from 0 to 13.5. In that case I get 243 π 4 which is also incorrect. I'm sure I've misinterpreted what it means to rotate the function about the axis x = 0.5
asked 2022-08-06
Finding the volume of O
Let K be a number field, r 1 denotes the number of real embeddings and 2 r 2 denotes so that r 1 + 2 r 2 = n = [ K : Q ]. Define the ring homomorphism, canonical imbedding of K, σ : K R r 1 × C r 2 as
σ ( x ) = ( σ 1 ( x ) , , σ r 1 ( x ) , , σ r 1 + r 2 ( x ) )
We only consider the 'first' r 2 complex embeddings since the rest are conjugate of the previous ones.
So, there is this lemma that I do not know how to start, what to think:
Lemma.If M is a free Z -module of K of rank n, and if ( x i ) 1 i n is a Z -base of M, then σ ( M ) is a lattice in R n , whose volume is v ( σ ( M ) ) = 2 r 2 | det 1 i , j n ( σ i ( x j ) ) | .
The determinant on the right hand side is Δ K as far as I know. However, I appreciate any help to understand the proof of this lemma.
Edit: The lemma does not say about anything about O , but it is a free Z -module of rank n. So the lemma is more general, O is a specific case but I wrote the title as this way. You can either take a Z -base for O and continue or work on any free Z -module of rank n.
asked 2022-07-16
Volume of Generalized Tetrahedron in R n .
I'm having difficulty finding the volume of a tetrahedron in R n .
Find the volume of a generalized tetrahedron in R n bounded by the coordinate hyperplanes and the hyperplane x 1 + x 2 + . . . + x n = 1
In two dimensions, we have 0 1 1 x 1 d x 1 . In three dimensions, I got something like 0 1 0 1 x 1 1 x 2 d x 2 d x 1 .
I am off to a good start?
asked 2022-09-11
The length of a cuboid is 60 cm. Its height is 40% of its length and 3\4 of its width. calculate the volume of the parallelepiped.
asked 2022-08-11
Finding a volume integral in an ellipsoid
I am trying to find the volume integral of ρ = ρ 0 ( R 2 r 2 R 2 ) inside an ellipsoid given by x 2 ( 3 R ) 2 + y 2 ( 4 R ) 2 + z 2 ( 5 R ) 2 = 1
I've tried using jacobian to move from an ellipsoid to an unit ball by these mapping relationships x = 3 R u, y = 4 R v, z = 5 R w.
But the resulting integral is still heavy
ρ 0 ( 1 ( 9 u 2 + 16 w 2 + 25 w 2 ) ) 60 d u d v d w
Does anyone have any insight to a more elegant way.

New questions