Solve Volume Problem: Find Region Bounded by Line & Curve

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In summary, the problem is to find the volume of a solid formed by the region bounded by the line y=x and the curve y=x^2, rotated about the line x=1. The two possible methods for finding the volume are washer and shell, with the latter potentially avoiding the use of a square root term in the integral. In the attempt at a solution, there was initially some confusion over the use of the square root and the need to square the radii. However, after revising the integral, the result was confirmed to be pi/6.
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Homework Statement



Find the volume of the soild formed.
The region bounded by the line y=x, and the curve y=x^2. about the line x=1

Homework Equations



Either Washer or Shell

The Attempt at a Solution



I want to use washer, but have to switch to dy.
I want to use shell, but don't think it will work.

I don't think this is right, but here is what I have.

integral of pie[(1-y^(1/2))-(1-y)]dy from 0 to 1.

Thanks, please help.
 
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  • #2
Ok for a washer the area is pie(r1^2 - r2^2), where r1 is the larger radius and r2 is the smaller radius. So I think you just need to take a closer look at which is the larger and smaller radius and remember to square them too.

OTOH if you used shells you might not have to deal with a square root term in your integral.
 
  • #3
the about x=1 is what mess me up. i meant to spuare all the R's but forgot, because this is hard to see.

integral of pie[(1-y)^2-(1-y^(1/2))^2]dy from 0 to 1.

this is right now?

I got pi/6, it looks right, but is it?
 
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  • #4
Looks correct to me.
 

FAQ: Solve Volume Problem: Find Region Bounded by Line & Curve

What is the formula for finding the volume of a region bounded by a line and a curve?

The formula for finding the volume of a region bounded by a line and a curve is to first determine the area of the cross-section of the region, which can be done by using the formula for finding the area under a curve. Then, multiply this area by the length or height of the region to find the volume.

What is the difference between finding the volume of a region bounded by a line and a curve and finding the area under a curve?

The main difference is that when finding the volume of a region bounded by a line and a curve, you are calculating the volume of a three-dimensional object. In contrast, when finding the area under a curve, you are calculating the area of a two-dimensional shape on the x-y plane.

What is the significance of finding the volume of a region bounded by a line and a curve?

Finding the volume of a region bounded by a line and a curve is important in many fields of science, such as physics, engineering, and chemistry. It allows us to calculate the amount of space occupied by a three-dimensional object, which is crucial in understanding and predicting its behavior.

What challenges might arise when solving volume problems for regions bounded by a line and a curve?

One challenge that may arise is determining the bounds of the region. This can be tricky if the line and curve intersect multiple times or if the region has irregular or complex shapes. Another challenge could be finding the area of the cross-section, which may involve using advanced mathematical techniques such as integration.

Are there any real-life applications of solving volume problems for regions bounded by a line and a curve?

Yes, there are many real-life applications of finding the volume of regions bounded by a line and a curve. For example, engineers may use this concept when designing structures such as bridges or buildings. Chemists may use it to calculate the volume of a reaction vessel or the amount of a substance needed for a reaction. It is also used in physics to determine the volume of an object or the amount of space it displaces.

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