Answer to Question #308964 in Calculus for Saifi

Question

Apply definition of antiderivative and find area under the curve of f(x) = x^1/2 between x=0 and

x=1

Answer 1

Solution

The required area is calculated as:

$A = \int\limits_a^b {f\left( x \right)\,} dx\$

Here $f(x)=\sqrt{x}$ and the limits are $a=0$ and $b=1$

Therefore, the required area is

$A = \int\limits_0^1 {\sqrt x \,} dx\$

$A=[\frac{x^\frac{3}{2})}{\frac{3}{2}}]_{0}^{1}$

$A = \frac{2}{3}\left[ {{x^{{\textstyle{2 \over 3}}}}} \right]_0^1\$

$A = \frac{2}{3}\left[ {{{\left( 1 \right)}^{{\textstyle{2 \over 3}}}} - {{\left( 0 \right)}^{{\textstyle{2 \over 3}}}}} \right]\$

$A = \frac{2}{3}\left[ {1 - 0} \right]\$

$A = \frac{2}{3}$ square units

The shaded area below shows the required area.