Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together. This is done for every point x, and in the end all the products are added to give the final area. That's it--that's how the area under the curve is found. Where did we use the tangent in this, that they should become so important in calculus?