Finding volume via integration

When we are asked to find the volume for example of a sphere, we slice it into numerous disks with small depth dr and get the infinitesimal volume dV. And then we sum those slices together by putting an integral symbol. But my question is the basic integration we learnt was to find area under the curve.In that case we proved that the area is simply the anti derivative of the function. But here how do we use integration by putting an integral symbol before Adr without proving that the derivative of the volume is the area of the disks? I am so disturbed by this as I didn't see any author giving the proof with proper intuition.

When we are asked to find the volume for example of a sphere, we slice it into numerous disks with small depth dr and get the infinitesimal volume dV. And then we sum those slices together by putting an integral symbol. But my question is the basic integration we learnt was to find area under the curve.In that case we proved that the area is simply the anti derivative of the function. But here how do we use integration by putting an integral symbol before Adr without proving that the derivative of the volume is the area of the disks? I am so disturbed by this as I didn't see any author giving the proof with proper intuition.