What is the Region Bounded by a Unit Circle in the yz Plane and x = 0?

[joemama69]

Homework Statement

evaluate the integral y²z²dV over W, which is the region bounded by x = 1 - y² - z² adn the plane x = 0

Homework Equations

The Attempt at a Solution

since x = 0, that makes y² + z² = 1, unit circle in the yz plane right?

so would the answer be the area of the circle times y²z²

 

[tiny-tim]

evaluate the integral y²z²dV over W, which is the region bounded by x = 1 - y² - z² adn the plane x = 0
…
so would the answer be the area of the circle times y²z²

uhh? the answer is a number … how can ∫∫∫ y²z²dV have any y or z in it?

 

[Cyosis]

Draw the region. You will see that it is a paraboloid with a maximum at y=z=0 and x=1. The paraboloid is cut off at x=0 so the region looks like the top of paraboloid. Which is not just a circle.

 

[Cyosis]

Those integration limits are not correct. For A) if your integration order is dzdydx there will be a z left in your answer. Every other order of integration has a similar problem so it can't be right.

For B) you don't integrate over x,y and z, but over r, theta and x. This means you've to find limits for r, theta and x. The polar coordinates in this case would be $x=x, y=r \cos \theta, z=r \sin \theta,y^2+z^2=r^2$. Try to find the correct boundaries for both A and B. Realize the region is a paraboloid with an extremum at 1 on the x-axis and cut off by the yz-plane.

I will leave it for you to decide which one is easier.

 

[HallsofIvy]

You might try first swapping x and z so that the integrand is $x^2y^2dV$ and the bounding surfaces are [math]z= 1- x^2- y^2[/math] and z= 0.

In cylindrical coordinates that would be $(r^2cos^(\theta)r^2sin^2(\theta) rdrd\theta dz= r^5 sin^2(\theta)cos^2(\theta)$ and the boundaries [math]z= 10- r^2[/math] and z= 0.

 

[joemama69]

Well I was originally thinking polar was easier but i think I've changed my mind.

BTW I am not using x²y² and z = 1 - x² - y², z = 0

I need help setting up my limits

z.. 0 to 1 - x² - y²

x... -(1-y²)^1/2 to 1-y²)^1/2

y... -1 to 1

whats right and what's wrong

 

[HallsofIvy]

So you are integrating over the region bounded by $z= 1- x^2- y^2$ and z= 0?

I would recommend converting to cylindrical coordinates: $x= r cos(\theta)$ and $y= r sin(\theta)$ so that $z= 1- r^2$. Don't forget that the "differential of volume" in cylindrical coordinates is $dV= r dr d\theta$.

If you are required to do this in rectangular coordinates (or just want to), then, yes, your limits of integration are correct. Projecting the parboloid onto the z= 0 plane, you get the circle $x^2+ y^2= 1$. y going from -1 to 1 will cover that and, for any given y, x will go from $-\sqrt{1- y^2}$ to $\sqrt{1-x^2}$. Finally, for any given x and y, z going from 0 to $1- x^2- y^2$ will cover the solid.

 

[joemama69]

Ok, i tried it in cylindrical but i got my limts wrong

z is from 0 to 1-r²

r is from 0 to $$\sqrt{1-z}$$

theta is from 0 to pi

 

[Cyosis]

You switched z and x around right? You can't have a z in your limits, it will mean you end up with a variable. The radii of the circular slices vary from the base r=1 to the top r=0. Also to integrate over the full circular slice you need to adjust your $\theta$ limits.

 

[joemama69]

im integrating over dz,dr,dQ Q = theta

dz = 0 to 1-r²

dr = 0 to 1

dQ = 0 to 2pi

$$\int\int\int$$r⁴cos²Q sin²Q r dz dr dQ

 

[Cyosis]

Yes that should give you the same answer as the Cartesian method.

 

[joemama69]

ok i got stuck integrating

in integrated z and pluged in the limits and now I am integrating r

$$\int$$(r⁵ - r⁷)cos²Q sin²Q dr = 1/24 ...

1/24 $$\int$$ cos²Q sin²Q dQ =

1/24 $$\int$$ ( - /sin²Q) sin²Q dQ =

1/24 $$\int$$ sin²Q - sin⁴Q dQ =

Now I am stuck... my calculator gave me pi/4, but how do i do it by hand

 

[HallsofIvy]

$sin^2(Q)- sin^4(Q)= sin^2Q(1- sin^2(Q))= sin^2(Q)cos^2(Q)$
Use trig identities $sin^2(Q)= (1/2)(1- cos(Q))$, $cos^2(Q)= (1/2)(1+ cos(Q))$. You may need to use those twice.

 

[joemama69]

i haven't those identities in a while

1/24 $$\int$$cos²Q sin²Q dQ =

1/24 $$\int$$(1/2 + 1/2 cosQ)(1/2 - 1/2 cosQ) dQ =

1/96 $$\int$$(1 + cos²Q) dQ =

1/96 $$\int$$(1 + 1/2 + 1/2 cosQ) dQ =

1/96 $$\int$$(3/2 + 1/2 cosQ) dQ =

1/96 (3Q/2 + 1/2 sinQ)] from 0 to 2pi = pi/32

my calc said it was pi/4 do u see any problems

 

[Cyosis]

I would personally start out with $\cos^2 \theta \sin^2 \theta=(\cos \theta \sin \theta)^2=\frac{1}{4}\sin^2 (2\theta)$. This way you can skip a few steps.

You used Halls identities which are incorrect the correct ones are $\cos^2x=\frac{1}{2}(1+\cos 2x),\sin^2x=\frac{1}{2}(1-\cos 2x)$. Note the double angle.

 

[joemama69]

ok when i integrated with the last identity i got pi/96 did u get this too

 

[Cyosis]

I got pi/96.