Find the volume of a described solid

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In summary, to find the volume of the solid with base x^2+y^2=9 and isosceles right triangle cross sections, use the formula (1/2)(\sqrt{9-x^2})(\sqrt{9-x^2}) and the limits of integration from 0 to 3 in the integral \pi\int 4.5-.5x^2 dx to get a final volume of 18\pi.
  • #1
Physicsisfun2005
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I don't have the answers in the back of my book and I really want to know if i did this correctly since its graded. the Problem is: Let the region bounded by [tex]x^2+y^2=9[/tex] be the base of a solid. Find the Volume if cross sections taken perpendicular to the base are isosceles right triangles.
i know a triangle is [tex].5bh[/tex] and the base of it will be [tex]\sqrt{9-x^2}[/tex] so for volume the final answer is [tex]\pi\int 4.5-.5x^2 dx[/tex] with the limits from -3 to 3 and i get [tex]18\pi[/tex] for volume.....is this right?
 
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  • #2


Hello,

Thank you for your question. It appears that you have correctly set up the integral to find the volume of the solid. However, I would recommend double checking your limits of integration. Since the base of the solid is a circle with radius 3, the limits of integration should be from 0 to 3 (not -3 to 3). This is because the solid will only exist in the positive x-direction.

Also, I would suggest using the formula for the area of an isosceles right triangle, which is (1/2)(base)(height) = (1/2)(\sqrt{9-x^2})(\sqrt{9-x^2}). This will give you a slightly different integral, which should still result in the same volume of 18\pi.

I hope this helps and good luck with your graded assignment! Remember to always double check your work and make sure your limits of integration are accurate.
 
  • #3


Your calculation looks correct to me! To double check, you can graph the region and the cross sections to make sure they match up with your calculations. Also, it never hurts to double check your integration and make sure you didn't make any mistakes. But overall, it looks like you have the right approach and answer. Great job!
 

What is the definition of volume?

Volume is the amount of space occupied by an object or a substance.

What is the formula for finding the volume of a solid?

The formula for finding the volume of a solid depends on the shape of the solid. For example, the formula for finding the volume of a cube is V = s^3, where s is the length of one side of the cube.

How do you measure the dimensions of a solid?

The dimensions of a solid can be measured using a ruler, caliper, or any other measuring tool. For irregular shapes, you can use displacement method by submerging the solid in water and measuring the amount of water displaced.

What is the unit of measurement for volume?

The unit of measurement for volume depends on the unit of measurement used for the dimensions of the solid. For example, if the dimensions are measured in centimeters, then the volume will be measured in cubic centimeters (cm^3).

Can the volume of a solid be negative?

No, the volume of a solid cannot be negative. Volume is a physical quantity and cannot have a negative value. If the calculated volume is negative, then there may be an error in the measurement or calculation process.

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