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If you rotate an isosceles triangle by 2pi you are generating the volume twice over as each half-triangle will make the whole solid.
Why do I get the wrong answer when using the shell method?
Now if you had a cone with the same base radius and height, the volume would be less, wouldn't it.
seems reasonable, doesn't I?
I can take you step by step through the calculus if you wish. That way you'll learn some calculus too.
anyone explain to me where does the 1/3 of the formula of the volume of the cone and pyramid come from?need the answer ryt away. like what if without the idea first of having 1/3bh for their volume..how will we know it. or lets just say how to prove volume of the cone v=1/3bh .please in an understnding and clear way and as much as possible not just calculus..simple way ..thankss so much
The volume of a right circular cone with radius r and height h, equals the
area of the right triangle (let the base = r and the height = h), which is
being revolved along the line containing the line segment h, multiplied by the
circumference using the r/3 part of the centroid* as the radius of revolution.
The centroid of a triangle is where all of its medians intersect.
The centroid is the geometric center of the triangle.**
Then the formula for the volume is
the area of the triangle, multiplied by the circumference at the
geometric center (centroid), and using r/3 as the radius of revolution.
- - . . - - . . - - . . - - . . - - . . - - . . - - . . - -
* Suppose a right triangle is situated on the xy-plane with the
radius extending from (0, 0) to (r, 0) and the height extending
from (0, 0) to (0, h).
The x-coordinate of this centroid is r/3. (This can be worked out using
So think of a section of the pyramid made by an area s with a thickness of delta h.
Use integration to add up all such volumes thus:
So for any right* pyramid the volume is one third the height times the base area.
*Right here means that the axis is at right angles to the base. It is relatively easy to adapt this proof for non right pyramids.
that dosnt make any sense
ok, this is how i'd do it:
Using Calculus, by integration, u need to arrive the volume of the cone
Can anyone help me, how to derive the formula of the volume of the cone 1/3*pi*r^2*h ?