Calculating the Volume of a Concave Lens Using Multiple Integration

In summary, the question is asking for the volume of a circular concave lens with two refracting surfaces described by z=1/2(x2+y2+1) and z=-1/2(x2+y2). The solution involves using multiple integration, specifically integrating "1" over the region in the xy plane covered by the lens. The area of a circle with radius 2 can be used to determine the volume. The second surface is the negative of the first, but the same method can be applied.
  • #1
gboff21
50
0

Homework Statement



A circular concave lens of radius 2 units, has two refracting surfaces described by z=1/2(x2+y2+1) and z=-1/2(x2+y2) What is the volume of the glass?


Homework Equations



Over a region R: V=[tex]\int\int\int dA[/tex]


The Attempt at a Solution



I have no idea how to start this. Polar coordinates, bounded regions I don't know anything apart from the fact that I need to use multiple integration.
 
Physics news on Phys.org
  • #2
A line from [itex]z= (1/2)(x^2+ y^2)[/itex] straight up (parallel to the z axis) to [itex]z= (1/2)(x^2+ y^2+ 1)[/itex] has length 1 so the volume is simply the integral of "1" over the region, in the xy plane, the lens covers. And that is just the 1 times the area of that region! What is the area of a circle of radius 2?
 
  • #3
The second surface is [itex]z= -(1/2)(x^2+ y^2+ 1)[/itex]. Its negative. Even so, I'm doing this question for an exam on monday out of practise for multiple integrals. So how would you do this using integration?
 

What is the definition of volume of a concave lens?

The volume of a concave lens refers to the amount of space that the lens takes up, measured in cubic units. It is the three-dimensional measurement of the lens, including its thickness and curvature.

How is the volume of a concave lens calculated?

The volume of a concave lens can be calculated using the formula V = πr²h, where V represents volume, π is approximately 3.14, r is the radius of the lens, and h is the height or thickness of the lens. This formula assumes that the lens is a perfect circle.

What factors affect the volume of a concave lens?

The volume of a concave lens is affected by its radius of curvature and its thickness. A lens with a smaller radius of curvature or a greater thickness will have a larger volume. Additionally, the type of material the lens is made of can also impact its volume.

How does the volume of a concave lens relate to its optical properties?

The volume of a concave lens does not directly affect its optical properties, such as its focal length or refractive index. However, a lens with a larger volume will typically have a thicker center and thinner edges, which can impact its optical properties.

Why is the volume of a concave lens important in optics?

The volume of a concave lens is important in optics because it helps determine the physical characteristics of the lens, such as its weight and size. This information is useful in the design and manufacturing of lenses for various optical applications. Additionally, knowing the volume of a lens can also help with calculating its magnification power.

Similar threads

  • Introductory Physics Homework Help
Replies
3
Views
822
  • Introductory Physics Homework Help
Replies
5
Views
4K
  • Introductory Physics Homework Help
Replies
1
Views
960
  • Introductory Physics Homework Help
Replies
7
Views
5K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
276
  • Introductory Physics Homework Help
Replies
9
Views
8K
Back
Top