Chapter 4 - Integrals - 4.3 The Fundamental Theorem of Calculus - 4.3 Exercises - Page 327: 24

3

Evaluate the integral: $\int^8_1 (x^{-\frac{2}{3}}) dx$ Recall the 2nd part of the Fundamental Theorem of Calculus: $\int_a^bf(x)dx = F(b) - F(a)$ Find $F(x)$: $F(x) =\int^8_1 (x^{-\frac{2}{3}}) dx$ $F(x) = \frac{3}{1}x^{\frac{1}{3}}$ $F(x) = 3\sqrt[3] x$ Now evaluate $F(b) - F(a)$: $F(8) - F(1)$ $3\sqrt[3] 8 -3\sqrt[3] 1$ $3(2) - 3(1)$ $6-3 = 3$