What Is the Surface Area of a Sphere Inside a Paraboloid?

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Homework Help Overview

The problem involves finding the surface area of a sphere defined by the equation x^2 + y^2 + (z-2)^2 = 4 that lies within a paraboloid described by z = x^2 + y^2. The context centers around the intersection of these two geometric shapes and the application of double integration to determine the area of the spherical cap.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the intersection of the sphere and paraboloid, with one attempting to replace variables in the equations to find limits for integration. There are questions about the correct parameters for integration and the specific equations used in the calculations. Some participants express uncertainty about the integration process and the results obtained.

Discussion Status

The discussion is ongoing, with participants providing hints and clarifications regarding the setup of the problem. There is recognition of the need to differentiate between the area of the sphere that lies inside the paraboloid and the area of the paraboloid itself. Multiple interpretations of the integration limits and methods are being explored.

Contextual Notes

Participants note the intersection point at z = 3 and the need to consider the height of the spherical cap. There is mention of using polar coordinates for integration, but uncertainty remains about the correct limits for r and theta.

jaredmt
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Homework Statement


Find the area of the sphere x^2 + y^2 + (z-2)^2 = 4 that lies inside paraboloid z = x^2 + y^2

Homework Equations


The Attempt at a Solution



when i take the equation of the spere and replace x^2 + y^2 with z i get: z(z-3) = 0
so they intersect at the plane z = 3.

were supposed to use the double integration rule. I am not sure what the parameters would be. when i convert to polar, i know feta is from 0 to 2pie. i tried making r from 0 to 3 but i got the wrong answer. idk whether i integraded the wrong equation or used the wrong parameters or both
 
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How are we supposed to know what you did wrong if you won't tell us what you did. What did you integrate? What did you get?
 
i don't know how to type the integration symbols

the double integral was ||Rx X Ry||
where x = x, y = y, z = x^2 + y^2

so basically (skipping some steps) the integral becomes: (1 + 4x^2 + 4y^2)^.5
which becomes: r(1 + 4r^2)^.5 when put in polar form
and like i said, i tried 0<feta<2pi and 0<r<3

i got like pi/6 * some quantity. but its supposed to be 4pi
 
It looks like you are finding the area of the paraboloid that lies in the sphere. The question asks for the area of the sphere that lies inside the paraboloid. It's the upper cap of the sphere.
 
You might also notice z=r^2 on the paraboloid. The limits for z are 0->3. The limits for r are not 0->3.
 
Hi jaredmt! :smile:

(have a theta: θ and a pi: π and a squared: ² and an integral: ∫)
jaredmt said:
when i take the equation of the spere and replace x^2 + y^2 with z i get: z(z-3) = 0
so they intersect at the plane z = 3.

That's right! :smile:

So you're now trying to find the area of a cap of a sphere of radius 2 from "height" 2 to "height" 1 (and you're told to use ∫∫).
were supposed to use the double integration rule. I am not sure what the parameters would be.

Hint: you can use either x and y parameters, or latitude and longitude parameters.

They both work (well, why wouldn't they? :rolleyes:), so you may as well try both of them! :smile:
 

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