Surface area enclosed by Cylinders

In summary, the surface areas enclosed by cylinders and cones can be calculated using calculus integration. For a region bounded by a cylinder, the volume is given by the integral of the difference between the top and bottom functions in cylindrical coordinates. For a region bounded by a cone, the volume is given by the integral of the height plus the radius over the cone's radius.
  • #1
RKD89
78
1
I want to know how Surface areas enclosed by Cylinders , Cones..etc can be calculated using calculus integration...
 
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  • #2
well a cylinder is just a square that 2 ends have been touched so wouldn't it be
2 pi r 2 + 2 pi r h or 2(pi r 2) + (2 pi r)* hand the inside and outside have the same surface area unless there's a thickness in which case R=distance to inner rim instead of outer this includes the top and bottom surface areas also
 
  • #3
RKD89 said:
I want to know how Surface areas enclosed by Cylinders , Cones..etc can be calculated using calculus integration...
In both of those, it would probably be best to use cylindrical coordinates.

If, for example, your region is bounded by the cylinder \(\displaystyle x^2+ y^2= R^2[/itex],
with top and bottom given by z= f(x,y) and z= g(x,y), respectively, then the volume is given by
[tex]\int\int (f(x,y)- g(x,y))dydc= \int_{r= 0}^R\int_{\theta= 0}^{2\pi} (f(r cos(\theta),r sin(\theta))- g(r cos(\theta),r sin(\theta))) r dr d\theta[/tex]

The volume of the region bounded above by the cone [itex]R^2(z-h)^2= x^2+ y^2[/tex] which, in cylindrical coordinates is [itex]R(z- h)= r[/itex], and below by z= 0, is given by
[tex]\int_{r= 0}^R\int_{\theta= 0}^{2\pi} z rdrd\theta= \int_{r= 0}^R\int_{\theta= 0}^{2\pi} h+ \frac{r}{R} rdrd\theta[/tex]\)
 

Related to Surface area enclosed by Cylinders

1. How do you calculate the surface area enclosed by a cylinder?

The surface area of a cylinder can be calculated by multiplying the circumference of the base by the height of the cylinder and adding the area of the two circular bases. The formula for surface area of a cylinder is 2πr(h + r), where r is the radius and h is the height.

2. What is the difference between lateral surface area and total surface area of a cylinder?

Lateral surface area refers to the area of the curved surface of the cylinder, while total surface area includes the area of the two circular bases in addition to the lateral surface area. The formula for lateral surface area is 2πrh, while the formula for total surface area is 2πr(h + r).

3. How does the surface area of a cylinder change when the radius or height is changed?

If the radius of a cylinder is increased, the surface area will also increase. Similarly, if the height of the cylinder is increased, the surface area will also increase. This is because both the radius and height are directly proportional to the surface area of a cylinder.

4. Can the surface area of a cylinder be negative?

No, the surface area of a cylinder cannot be negative. Surface area is a measurement of the total area of an object, and it cannot have a negative value.

5. How is the surface area of a cylinder related to its volume?

The surface area of a cylinder and its volume are not directly related. The volume of a cylinder is calculated by multiplying the area of the base by the height, while the surface area is calculated using the circumference of the base and the height. However, both the surface area and volume of a cylinder can be affected by changes in its dimensions.

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