Answer
a) $P(x \geq 3) \approx 0.268$
b) $P(x \leq 2)\approx 0.414$
c) $P(2 \lt x \lt 3)\approx 0.3178$
Work Step by Step
We are given $f(t)=\frac{1}{2\sqrt x}$ for $x$ in $[1,4]$
a) $P(x \geq 3)=\int^{4}_3\frac{1}{2\sqrt x}dx=\frac{1}{2}\int^{4}_3\frac{1}{\sqrt x}dx=x^{\frac{1}{2}}|^4_3 \approx 0.268$
b) $P(x \leq 2)=\int^{2}_1\frac{1}{2\sqrt x}dx=x^{\frac{1}{2}}|^2_1 \approx 0.414 $
c. $P(2 \lt x \lt 3)=\int^{3}_2\frac{1}{2\sqrt x}dx=x^{\frac{1}{2}}|^3_2 \approx 0.3178 $
