Question

In Problems 17 to 30, for the curve $\mathbf{y}\mathbf{=}\sqrt{\mathbf{x}}$, between $\mathit{x}\mathbf{=}\mathbf{0}$and $\mathit{x}\mathbf{=}\mathbf{2}$,

find:

The area under the curve.

Short answer

The area under the curve is obtained as $A=\frac{4\sqrt{2}}{3}$units.

Step-by-step solution

 Step 1: Definition of Area under the curve

The region is circumscribed by the function that is dealing with. In this, vertical lines denote the function's boundaries, and the x-axis is known as the area under the curve.

Representation of the area bounded by the curve

Draw the bounded region for the curve $y=\sqrt{x}$, between$x=0$and $x=2$.

Determination of area of the curve

Determine the value of area of curve.

$$\begin{gathered} A={\int }_0^2\sqrt{x}dx \\ ={\overline{)\frac{x^{1+{\displaystyle \frac{1}{2}}}}{1+{\displaystyle \frac{1}{2}}}}}_0^{2}n \\ =\frac{2}{3}{\left[x^{\frac{3}{2}}\right]}_0^2 \end{gathered}$$

Thus, the area under the curve is obtained as $A=\frac{4\sqrt{2}}{3}$units.