Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 5 - Integration - 5.3 Fundamental Theorem of Calculus - 5.3 Exercises: 91

Answer

$$\frac{{45}}{4}$$

Work Step by Step

$$\eqalign{ & \int_1^8 {\root 3 \of y } dy \cr & {\text{use property of radicals }}\root n \of {{a^m}} = {a^{m/n}} \cr & = \int_1^8 {{y^{1/3}}} dy \cr & {\text{integrate using the power rule for integration}} \cr & = \left. {\left( {\frac{{{y^{4/3}}}}{{4/3}}} \right)} \right|_1^8 \cr & = \frac{3}{4}\left. {\left( {{y^{4/3}}} \right)} \right|_1^8 \cr & {\text{using the fundamental theorem}} \cr & = \frac{3}{4}\left( {{{\left( 8 \right)}^{4/3}} - {{\left( 1 \right)}^{4/3}}} \right) \cr & {\text{simplify}} \cr & = \frac{3}{4}\left( {16 - 1} \right) \cr & = \frac{{45}}{4} \cr} $$
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