Area of Solid/Convert to Cylindrical and Spherical

In summary, the conversation discusses converting integrals from Cartesian to cylindrical coordinates, with an attempt at finding the volume resulting in an answer of 0. The expert summarizer notes that the work is correct and the misunderstanding lies in the calculation of an area instead of the intended integral.
  • #1
matt19weezy
2
0

Homework Statement


Convert to Cylindrical Coordinates from Cartesian

1st int(-2 to 2), 2nd int(-sqrt(4-x2) to sqrt (4-x2)), 3rd int((x2+y2) to 4) X dz dy dx.

I changed the integrals to Cylindrical 1st int(0 to pi), 2nd int(-2 to 2), 3rd int(r2 to 4) and the X to r cos(theta). r dz dr dtheta. I know this is wrong because I keep getting zero for the answer to area.


Homework Equations


Cylindrical- x=rcos(theta) y= r sin(theta) z=z

Cyl to Spherical- r= psin(fi) z=pcos(fi) fi=fi


The Attempt at a Solution


I changed the integrals to Cylindrical 1st int(0 to pi), 2nd int(-2 to 2), 3rd int(r2 to 4) and the X to r cos(theta). r dz dr dtheta. I know this is wrong because I keep getting sin(theta) and ending up with an area of 0.
 
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  • #2
But you aren't calculating area. I suppose you meant you are calculating the volume and it can't be zero. But you aren't calculating volume either or the integrand would be 1 instead of x. In fact your integrand is antisymmetric in x (an odd function) so you would expect an answer of 0. In short, your work is correct. It's just your understanding of what you are calculating that is wrong.
 
  • #3
Wow. Duh. Thank you so much. I have been thinking this was incorrect the whole day. Of course now I see that I'm not solving for an area, I'm solving the integral. Oh thank you so much
 

FAQ: Area of Solid/Convert to Cylindrical and Spherical

1. What is the formula for finding the area of a solid?

The formula for finding the area of a solid varies depending on the shape of the solid. For a cylindrical solid, the formula is A = 2πrh + 2πr^2, where r is the radius and h is the height. For a spherical solid, the formula is A = 4πr^2, where r is the radius. There are also specific formulas for other shapes such as cubes, pyramids, and cones.

2. How do you convert a solid to a cylindrical shape?

To convert a solid to a cylindrical shape, you will need to know the dimensions of the solid and use the formula V = πr^2h to calculate the volume of the solid. Then, you can use the formula A = 2πrh + 2πr^2 to find the surface area of the solid. Make sure to substitute the appropriate values for r and h in the formula.

3. How do you convert a solid to a spherical shape?

To convert a solid to a spherical shape, you will need to know the dimensions of the solid and use the formula V = (4/3)πr^3 to calculate the volume of the solid. Then, you can use the formula A = 4πr^2 to find the surface area of the solid. Make sure to substitute the appropriate value for r in the formula.

4. What is the difference between a cylindrical and spherical solid?

A cylindrical solid has a circular base and a curved surface, while a spherical solid has a curved surface and no base. Cylindrical solids also have a height, while spherical solids do not. Additionally, the formulas for finding the volume and surface area are different for each shape, as mentioned in the previous questions.

5. How can knowing the area of a solid be useful in real life?

Knowing the area of a solid can be useful in many real-life situations, such as calculating the amount of paint needed to cover a cylindrical or spherical object, determining the amount of material needed to create a specific shape, or understanding the capacity of a container. It is also a fundamental concept in geometry and calculus, which are widely used in fields such as engineering, architecture, and physics.

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