Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 4 - Integrals - 4.3 The Fundamental Theorem of Calculus - 4.3 Exercises: 32

Answer

$1 - \frac{1}{\sqrt{3}}$

Work Step by Step

Evaluate the Integral: $\int^{\pi/3}_{\pi/4}csc^2(θ)dθ $ Recall the 2nd part of the Fundamental Theorem of Calculus: $∫^b_af(x)dx=F(b)−F(a)$ Find $F(x)$: $F(x) =\int csc^2(x)dx$ $F(x) =-cot(x)$ Now Evaluate: $F(b) - F(a)$ $F(\frac{\pi}{3})-F(\frac{\pi}{4})$ $-cot(\frac{\pi}{3}) - -cot(\frac{\pi}{4})$ $-\frac{cos(\frac{\pi}{3})}{sin(\frac{\pi}{3})} + \frac{cos(\frac{\pi}{4})}{sin(\frac{\pi}{4})}$ $-\frac{\frac{1}{2}}{\frac{\sqrt3}{2}} + 1$ $1 - \frac{1}{\sqrt 3}$
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