Applications of Integration

Question

Find, by slicing, the volume of a cone whose height is 3 cm and whose base radius is 3 cm. Slice the cone as shown in the figure below.

Answer 1

I can't see your figure but I'm going to assume the cone is vertical with the base at the bottom. Let y be the vertical direction and x be the horizontal direction. The volume of the slice is:

dV = π·x²·dy

where x is the radius of the slice and dy is its thickness. Since we're integrating wrt y, we want to express the radius x in terms of y. The radius of the slice can be found by equating similar triangles in the cone:

(Height of Cone) / (Radius of Cone) = (Height from slice to top of cone) / (Radius of slice)

3/3 = (3-y)/x

x = 3-y

dV = π·(3-y)²·dy

Integrate from y = 0 to y = 3. Don't forget the units, cm³. From Geometry, we know the volume of a cone is π·r²·h/3. Check your answer against this formula for h = 3 and r = 3,