Use the method of Example 4a or Equation (1) to evaluate the definite integral.

Example 4a

Compute ∫₀^bx dx and find the area A under y = x over the interval [0, b], b > 0.

To compute the definite integral as the limit of Riemann sums, we calculate lim-0 =1f(c) Ax for partitions whose norms go to zero. Theorem 1 tells us that it does not matter how we choose the partitions or the points c as long as the norms ap- proach zero. All choices give the exact same limit. So we consider the partition P that subdivides the interval [0, b] into n subintervals of equal width Ax = (b 0)/n = b/n, and we choose c to be the right endpoint in each subinterval. The partition is = P = 0. b 2b 3b n' n' n' .. nb 'n n kb and Ck = So n k=1 f(c) Ax kb b n n = = = = n = k=1 n 6 n 6 2 kb n n k n(n + 1) 2 1 + f(c) = Ck Constant Multiple Rule Sum of First n Integers