Surface Area Problem: Find Area of Paraboloid Cut by Plane y=25

In summary, the conversation is about finding the area of the finite part of a paraboloid cut off by a plane. The method discussed involves using polar coordinates and integrating the function (1+4r^2) over a specific range of values for r and θ. The person asking for help wonders if they are doing something wrong and also asks about projecting the surface onto the x-z plane.
  • #1
bodensee9
178
0

Homework Statement


I am wondering if someone could help me with the following? I am supposed to find the area of the finite part of the paraboloid y = x^2+z^2 that's cut off by the plane y = 25. Now, wouldn't this be the same as the paraboloid z = x^2+y^2 that's cut off by the plane z = 25?

So, if that's right, then the surface area is ∫∫√((dz/dx)^2+(dz/dy)^2+1).
This seems easier to do in polar coordinates, so we basically have the following:

∫∫(1+4r^2)? And, r would be from 0 to 5, and 0≤θ≤2*pi?

Am I doing something wrong here? Or, how do I "project" the surface unto the x-z plane? Thanks!
 
Physics news on Phys.org
  • #2
Why do you think you are doing something wrong? Well done.
 

FAQ: Surface Area Problem: Find Area of Paraboloid Cut by Plane y=25

1) What is the formula for finding the surface area of a paraboloid cut by a plane?

The formula for finding the surface area of a paraboloid cut by a plane y=25 is: SA = 2π∫(y)(√(1 + (dy/dx)^2))dx, where y is the equation of the parabola and dy/dx is the derivative of y with respect to x.

2) How do you determine the boundaries for the integral when finding the surface area of a paraboloid cut by a plane?

The boundaries for the integral are determined by the points of intersection between the parabola and the plane y=25. These points can be found by setting the equations of the parabola and the plane equal to each other and solving for the x-values.

3) Can you use the same formula to find the surface area of any paraboloid cut by a plane?

Yes, the same formula can be used to find the surface area of any paraboloid cut by a plane. However, the boundaries for the integral and the equation for the parabola may vary depending on the specific paraboloid and plane being considered.

4) How do you interpret the result of the surface area calculation for a paraboloid cut by a plane?

The result of the surface area calculation represents the total area of the surface of the paraboloid that is above the plane y=25. This can be thought of as the area of a curved "cap" on top of the paraboloid.

5) Are there any practical applications for finding the surface area of a paraboloid cut by a plane?

Yes, there are several practical applications for finding the surface area of a paraboloid cut by a plane. For example, this calculation can be used in engineering and construction to determine the surface area of curved structures such as domes and arches. It can also be used in physics and astronomy to calculate the surface area of curved objects in space.

Similar threads

Find the dimensions that will minimize the surface area of a Rectangle
Replies
1
Views
892
Surface area of a circular cylinder cut by a slanted plane
Replies
10
Views
2K
Surface area bounded by 2 different planes
Replies
4
Views
1K
Minimal surface area for a fixed volume
Replies
5
Views
2K
Calculus of Variations, Isoperimetric, given surface area max volume
Replies
11
Views
3K
Understanding the argument of the surface area integral
Replies
7
Views
2K
Find the equation of the tangent plane and normal to a surface
Replies
1
Views
858
Element of surface area in spherical coordinates
Replies
7
Views
4K
Surface area of a shifted sphere in spherical coordinates
Replies
3
Views
3K
Find all points where surface normal is perpendicular to plane
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top