Volume of Solid w/ Semicircular Cross Sections in 1st Quadrant

[radtad]

The base od a solid is a region in the 1st quadrant bounded by the x-axis, y-axis and the line x+2y=8. If cross sections of the solidperpendicular to the x-axis are semicircles, what is the volume of the solid?

How come the answer isn't just the intgegral from 0-8 of 1/2pi(4-x/2)^2

 

[learningphysics]

The base od a solid is a region in the 1st quadrant bounded by the x-axis, y-axis and the line x+2y=8. If cross sections of the solidperpendicular to the x-axis are semicircles, what is the volume of the solid?

How come the answer isn't just the intgegral from 0-8 of 1/2pi(4-x/2)^2

That's what it looks like to me. Is it the wrong answer?

 

[radtad]

yea according to the answer key its wrong

 

[radtad]

choices are
a. 12.566 b. 14.661 c. 16.755 d 67.021 e 134.041

i keep ending up with choice d but the answer key says its choice c

 

[learningphysics]

Oh I see now. I misunderstood the problem. I did it by finding the solid after rotating about the x-axis and integrating... But the cross section of this solid is the circle whose diameter is from y=0 to y=4-x/2. We were thinking the cross section was a circle with diameter from y=-(4-x/2) to y=4-x/2. But if we make the radius (4-x/2)/2, then we get the right answer.

0-8 of 1/2pi[1/2(4-x/2)]^2 is correct.

 

[radtad]

yea i realized that too thanks