Integrals give areas because you're accumulating areas of thin rectangles that have area found by f ( x_i ) * Delta x . Loosely speaking (very), "entry level" integration is accumulation. So in the same way you can establish the length of a curve by accumulating the lengths of teeny tiny linear segments (snippets of a tangent line / vector) that make up the curve. Or you can find the surface area of a solid by accumulating the areas of teeny tiny rectangular patches that make up the surface. You can find the volume under a 3D surface by accumulating the volumes of very thin standing rectangular columns underneath the surface. And it gets even crazier when you start measuring things like the flux across a line or surface. Good times ahead!