The base of a certain solid is an equilateral triangle with altitude 11. Cross-s

Question

The base of a certain solid is an equilateral triangle with altitude 11. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula

V=baA(x)dx

applied to the picture shown above (click for a better view), with the left vertex of the triangle at the origin and the given altitude along the x-axis.

The lower limit of integration is a =  
The upper limit of integration is b =  
The diameter 2r of the semicircular cross-section is the following function of x:  
A(x)=  
Thus the volume of the solid is V =

Explanation / Answer

The lower limit of integration is a = 0
The upper limit of integration is b = 11
The diameter 2r of the semicircular cross-section is the following function of x: 2x/sqrt3
A(x)= pi (x^2)/6
Thus the volume of the solid is V = integral[0 to 11] pi (x^2)/6 dx

=[0 to 11] pi (x^3)/18 +c

=pi (11^3)/18 -0

=232.3

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