# The Area Bounded by the Curve y=x^2+1 from x=2 to x=3

In this tutorial we shall find the area of the region between the x-axis and the curve $y = {x^2} + 1$ from $x = 2$ to $x = 3$.

The graph of the function $y = {x^2} + 1$ is shown in the given diagram.

The required area of the shaded region is given by the integral of the form
$A = \int\limits_2^3 {ydx}$
$\begin{gathered} A = \int\limits_2^3 {\left( {{x^2} + 1} \right)dx} \\ \Rightarrow A = \left| {\frac{{{x^3}}}{3} + x} \right|_2^3 = \left( {\frac{{{3^3}}}{3} + 3} \right) – \left( {\frac{{{2^3}}}{3} + 2} \right) \\ \Rightarrow A = 9 + 3 – \frac{8}{3} – 2 = 10 – \frac{8}{3} \\ \end{gathered}$
$Area = \frac{{22}}{3}$

Which shows that the area under the curve.