Riemann Sums: Foundation of Integration

Draw a smooth curve representing a polynomial function that rises from left to right on a coordinate plane. Mark an interval along the x-axis from a starting point to an ending point. Divide this interval into n equal subintervals and draw vertical lines at each division point reaching up to intersect the curve. Below the curve, fill rectangular strips with equal widths spanning each subinterval, where the height of each rectangle corresponds to the function value at either the left or right endpoint of that subinterval. These rectangles form a step-like pattern that approximates the area under the curve, with the lower rectangles using left endpoints and upper rectangles using right endpoints, visually demonstrating the Riemann sum concept.