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 given function y = {x^2} + 1 as shown in the given diagram. The required area of the shaded region.


area-bounded-curve-2-3

The required area 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.