Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.8 Probability and Integration - Exercises - Page 448: 2


$C=\dfrac{3}{32}$and $P(3 \leq X \leq 4)=\dfrac{5}{32}$

Work Step by Step

We are given that $p(x)=Cx(4-x)$ $\int_0^{4}Cx(4-x) dx =\dfrac{4Cx^2}{2}-\dfrac{Cx^3}{3}]_0^4\\=\dfrac{32 C}{3}$ The condition will be satisfied when $\int_0^{4}Cx(4-x) dx=\dfrac{32C}{3}=1$ or, $C=\dfrac{3}{32}$ We will find $P(3 \leq X \leq 4)=\int_3^4 \dfrac{3}{32} x(4-x) dx \\=\dfrac{3}{32}[2x^2-\dfrac{x^3}{3}]_3^4\\=\dfrac{5}{32}$
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