Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.1 Exercises - Page 512: 48

Answer

$$y = \frac{1}{2}\arcsin \left( {\frac{{x - 2}}{2}} \right) + \frac{1}{2}$$

Work Step by Step

$$\eqalign{ & \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {4x - {x^2}} }} \cr & {\text{Separate the variables}} \cr & dy = \frac{1}{{\sqrt {4x - {x^2}} }}dx \cr & {\text{Complete the square}} \cr & dy = \frac{1}{{\sqrt {4 - \left( {{x^2} - 4x + 4} \right)} }}dx \cr & dy = \frac{1}{{\sqrt {4 - {{\left( {x - 2} \right)}^2}} }}dx \cr & {\text{Integrate both sides}} \cr & y = \frac{1}{2}\arcsin \left( {\frac{{x - 2}}{2}} \right) + C,{\text{ }}\left( {\bf{1}} \right) \cr & {\text{Use the initial condition }}\left( {2,\frac{1}{2}} \right) \cr & \frac{1}{2} = \frac{1}{2}\arcsin \left( {\frac{{2 - 2}}{2}} \right) + C \cr & C = \frac{1}{2} \cr & {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr & y = \frac{1}{2}\arcsin \left( {\frac{{x - 2}}{2}} \right) + \frac{1}{2} \cr & \cr & {\text{Graph}} \cr} $$
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