# 7.2 The Computation of the Definite Integral # Art Belmonte, Summer 1996 # These are the shaded gray examples which appear in Section 7.2. # In-line plots have been removed to conserve disk space. > restart; f:=x->x^2; 2 f := x -> x > dx:=(b - a) / n; b - a dx := ----- n > Sum(f(a + i*dx) * dx, i=1..n); n / i (b - a)\2 ----- |a + ---------| (b - a) \ \ n / ) ------------------------ / n ----- i = 1 > area:=Sum(f(a + i*dx) * dx, i=1..n); value("); n / i (b - a)\2 ----- |a + ---------| (b - a) \ \ n / area := ) ------------------------ / n ----- i = 1 2 3 2 2 2 a (n + 1) b a (n + 1) a b (n + 1) a b (n + 1) ------------ - ---------- + ------------- - ------------ n n 2 2 n n 2 2 2 3 2 3 a b (n + 1) a b (n + 1) a (n + 1) a (n + 1) - 2 ------------- + 2 ------------ + ----------- - ---------- 2 2 2 2 n n n n 3 3 3 2 3 b (n + 1) b (n + 1) b (n + 1) + 1/3 ----------- - 1/2 ----------- + 1/6 ---------- 3 3 3 n n n 2 3 2 2 2 b a (n + 1) b a (n + 1) b a (n + 1) - ------------- + 3/2 ------------- - 1/2 ------------ 3 3 3 n n n 2 3 2 2 2 b a (n + 1) b a (n + 1) b a (n + 1) + ------------- - 3/2 ------------- + 1/2 ------------ 3 3 3 n n n 3 3 3 2 3 2 a (n + 1) a (n + 1) a (n + 1) a b - 1/3 ----------- + 1/2 ----------- - 1/6 ---------- - ---- 3 3 3 n n n n 3 a + ---- n > Limit(area, n=infinity); value("); n / i (b - a)\2 ----- |a + ---------| (b - a) \ \ n / lim ) ------------------------ n -> infinity / n ----- i = 1 3 3 - 1/3 a + 1/3 b > Int(f(x), x=a..b); b / | 2 | x dx | / a > Int(f(x), x=a..b); value("); b / | 2 | x dx | / a 3 3 - 1/3 a + 1/3 b > Int(f(x), x); value("); / | 2 | x dx | / 3 1/3 x > Int(sqrt(x^5 + 1), x=1..2); evalf("); 2 / | 5 1/2 | (x + 1) dx | / 1 3.147104357 >