(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let r be a positive constant. Consider the cylinder x^{2}+ y^{2}≤ r^{2}, and let C be the part of the cylinder that satisfies

0 ≤ z ≤ y.

(1) Consider the cross section of C by the plane x = t (−r ≤ t ≤ r), and express its area in terms of r, t.

(2) Calculate the volume of C, and express it in terms of r.

(3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point (r cos θ, r sin θ, 0)

(0 ≤ θ ≤ π). Let b be the length of the line segment from the point (r cos θ, r sin θ, 0) to

the point (r cos θ, r sin θ, r sin θ). Express a and b in terms of r, θ.

(4) Calculate the area of the side of C with x^{2}+y^{2}= r^{2}, and express it in terms of r.

2. Relevant equations

Not sure

3. The attempt at a solution

Let the x - axis horizontal, y - axis vertical and z - axis in / out of page. I imagine there is circle on xy plane with radius r then it extends out of page (I take out of page as z^{+}) to form 3 D cylinder.

(1) I image there is rectangular plane that cuts the cylinder and the shape of the cross section is rectangle. The length of the triangle is 2y and the width is z. Taking z = y, the area will be 2y^{2}= 2 (r^{2}- x^{2}) = 2 (r^{2}- t^{2})

But the answer is 1/2 (r^{2}- t^{2})

(2) The volume of cylinder = base area x height = πr^{2}. z and by taking z = y = r I get πr^{3}but the answer is 2/3 r^{3}

(3) I get this part

(4) I am not sure what "area of side of C" is. Is it surface area of the circular part of cylinder?

Thanks

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# Homework Help: Cylinder in 3 D

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