Chapter 4 - Integrals - 4.3 The Fundamental Theorem of Calculus - 4.3 Exercises - Page 327: 22

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Evaluate the integral: $\int^1_0 (1 -8v^3 +16v^7) dv$ Recall the 2nd part of the Fundamental Theorem of Calculus: $\int_a^bf(x)dx = F(b) - F(a)$ Find $F(v)$: $F(v) = \int^1_0 (1 -8v^3 +16v^7) dv $ $F(v) =v - \frac{8}{4}v^4 +\frac{16}{8}v^8$ $F(v) =v - 2v^4 +2v^8$ Now evaluate $F(b) - F(a)$: $F(1) - F(0)$ $(1 - 2(1)^4 +2(1)^8 )- (0 - 2(0)^4 +2(0)^8) $ $(1-2+2) - 0=1$