why is the cylinder's volume = pi . r^2. h?

Question

you know in calculus books when they want to estimate the volume of a shape in an interval like [a,b] they devide this interval into too many suintevals. then there will be too many rectangles. then they rotate each rectangle. and then they use cylinder's volume as a defined volume to find the total volume. (like an axiom)
my teacher said that in fact it's proven why the volume of a cylinder is that.
can you help me find it?

Answer 1

Accepting the fact that :-
V = Area of base x height
Now derive formula for area of circular base of radius r :-
Consider a thin concentric strip of width δx whose radius is x.
Area of strip = (2πx) (δx)
Area of base = ∫ 2πx dx between lims. 0 and r
Area of base = 2π x²/2 between lims 0 and r
Area of base = A = π r ²
V = A h
V = (π r ²) h

Answer 2

Any solid figure that has two bases with same area and are joined by a curved plane which makes a 90 degree angle with the base has the volume as
area of base* height
So the volume of a right cylinder will be area of the base*height
So area of base=pi*r^2
So the volume is pi*r^2*h.

Answer 3

Picture a circle/ the area is pi r^2
Place another circle on top of it. And keep going. You are increasing the height. Now we call it volume and the volume is the base area times how ever high it is. this is true of all rectangular prisms, too.