Consider {{MATH_187}}. Approximate the area between the graph of {{MATH_188}} and the {{MATH_189}}-axis on the interval {{MATH_190}} using a {{CMD_191}}midpoint {{CMD_192}}Riemann sum with {{MATH_193}} rectangles. {{CMD_194}} First note that the width of each rectangle is {{MATH_195}} The grid points define the edges of the rectangle and are seen below: {{CMD_196}} {{ENV_197}} {{CMD_198}} On the other hand, the sample points identify which endpoints we use: {{CMD_199}} {{ENV_200}} {{CMD_201}} It is helpful to collect all of this data into a table: {{MATH_202}} Now we may write a midpoint Riemann sum {{MATH_203}} which evaluates to {{ENV_204}} and we find {{MATH_205}} {{CMD_206}} {{CMD_207}}