(4) \\int_{1}^{9} \\sqrt{x} dx
(4) \\int_{1}^{9} \\sqrt{x} dx
Answer
Explanation:
Step1: Find the antiderivative
The antiderivative of (x^{\frac{1}{2}}) is (\frac{2}{3}x^{\frac{3}{2}}) (using the power rule (\int x^n dx=\frac{x^{n + 1}}{n+1}+C), here (n=\frac{1}{2})).
Step2: Apply the fundamental theorem of calculus
[ \begin{align*} \int_{1}^{9}\sqrt{x}dx&=\left[\frac{2}{3}x^{\frac{3}{2}}\right]_{1}^{9}\ &=\frac{2}{3}(9)^{\frac{3}{2}}-\frac{2}{3}(1)^{\frac{3}{2}} \end{align*} ]
Step3: Simplify the expression
Since (9^{\frac{3}{2}}=(3^2)^{\frac{3}{2}} = 3^3=27) and (1^{\frac{3}{2}} = 1), then (\frac{2}{3}(27)-\frac{2}{3}(1)=\frac{54 - 2}{3}=\frac{52}{3})
Answer:
(\frac{52}{3})