Application of Integration [Rectangles under Graph]

The graph of \(y={{\mathrm{e}}^{x}}\) is shown in the diagram. Rectangles, each of width \(\frac{1}{n}\), where \(n\) is an Integer, are drawn under the curve.

1. The region \({{A}_{1}}\) is bounded by the \(x\)- axis, the \(y\)- axis, the line \(x=1\) and the curve \(y={{\mathrm{e}}^{x}}\).
   The area of \({{A}_{1}}\) may be appropriated by the total area of all the \(n\) rectangles.
   Given that \(1+x+{{x}^{2}}+{{x}^{3}}+...+{{x}^{n}}=\frac{1-{{x}^{n+1}}}{1-x}\), show that the area of \({{A}_{1}}\) is approximately \(\frac{\mathrm{e}-1}{n\left( {{\mathrm{e}}^{\frac{1}{n}}}-1 \right)}\).[3]
2. The region \({{A}_{2}}\) is bounded by the \(y\)- axis, the line \(y=\mathrm{e}\) and the curve \(y={{\mathrm{e}}^{x}}\). Using (i), explain how the area of \({{A}_{2}}\) can be approximated and state, with a reason, whether this area is an underestimation or an overestimation.[2]