Sketch the line y = (r/h)*x, in the x-y plane, where r and h are arbitrary constants representing the radius and height of the cone. Sketch the result of taking the line segment from x = 0 to x = h and rotating that line 360 degrees around the x-axis, which is a cone.
Draw vertical lines through the cone to indicate that the cone is sliced into many disks. Write down the area of an infinitely thin version of one of these disks, which would be a circle. You should get Area = pi*R^2, where R is the radius of that particular circle or disk.
Write an integral representing this situation. You need to go in an x direction to accumulate all of your vertical slices into a cone, so write the integration with respect to x. The integration starts at 0 and ends at h, so put these as your endpoints. The area of each slice is pi*R^2, so put this as your function. Put this all together and get:
S(0,h) pi*R^2 dx
where S(0,h) represents the integral symbol with endpoints 0 and h.
Change R into a function of x so you can integrate with respect to x. Since R is always the same as y, which equals (r/h)*x, substitute (r/h)*x for R. You should now have:
S(0,h) pi*((r/h)*x)^2 dx
Integrate the integral S(0,h) pi*((r/h)*x)^2 dx. Square the function on the inside to get S(0,h) pi*(r^2/h^2)*x^2 dx. Using the power rule, change the x^2 to (1/3)*x^3, then write all the constants in front. You get (1/3)*pi*(r^2/h^2)*x^3 evaluated from 0 to h, which is (1/3)*pi*(r^2*h^3/h^2) - 0. Simplify the expression, and you have (1/3)*pi*r^2*h, which is the same as the formula for the volume of a cone.