Volume of solid having triangle base and semicircle slice

songoku
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Homework Statement
Calculate the volume of solid having triangle base with vertices (0, 0) , (2, 0) and (0, 3) whose slice perpendicular to x-axis is semicircle
Relevant Equations
Volume = ##\int_p^{q} A(x) dx##
First, I tried to find the equation of line passing through (2, 0) and (0, 3) and I got ##y=3-\frac{3}{2}x##

Then I set up equation for the area of one slice, ##A(x)##
$$A(x)=\frac{1}{2} \pi r^2$$
$$=\frac{1}{2} \pi \left( \frac{1}{2}y\right)^2$$
$$=\frac{1}{2} \pi \left(\frac{3}{2}-\frac{3}{4}x \right)^2$$

Am I correct until this point? Thanks
 
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I am not sure I got right image of the solid. Could you show me your sketch ?
 
anuttarasammyak said:
I am not sure I got right image of the solid. Could you show me your sketch ?
1643446320549.png

Hopefully it is clear enough

Thanks
 
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Thanks. Clear enough. So we observe the volume of the solid is half the volume of the cone which has base of area ##(3/2)^2\pi## and height 2. It would be beneficial for you to check answer.

Your nice sketch leads me to
V=\frac{1}{2}\int_0^2 \pi [\frac{3- \frac{3}{2}x}{2}]^2 dx
 
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anuttarasammyak said:
Thanks. Clear enough. So we observe the volume of the solid is half the volume of the cone which has base of area ##(3/2)^2\pi## and height 2. It would be beneficial for you to check answer.

Your nice sketch leads me to
V=\frac{1}{2}\int_0^2 \pi [\frac{3- \frac{3}{2}x}{2}]^2 dx
I also get the same result but the solution is:
$$V=\int_{0}^{2} \frac{1}{2} \left(\pi \left(\frac{1}{2}-\frac{x}{4}\right)^2\right)dx$$

That's why I am confused at which part I got it wrong, but I think maybe the solution is not correct

Thank you very much anuttarasammyak
 
The answer you referred seems to set vertexes are (0,0),(2,0), and (0,1), not (0,3).
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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