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[tex]2 \pi r[/tex]

i get

[tex]\int 2 \pi r dr=\pi r^2[/tex]

i figured its because the area of a circle can be seen as the sum of circumference's of circles with radius 0 to radius [tex]r[/tex]

i was thinking if the half volume of a ball also be seen as made from the sum of areas of circules with radius 0 to radius [tex]r[/tex]?

integrating [tex]\pi r^2[/tex] gives [tex]\pi \frac{r^3}{3}[/tex]

so multiplying by two should give the ball volume formula but it does not why is that?