Question

Area bounded by $y = {e^x}$ and lines x=0 and y=e is given by:
(This question has multiple correct options)
A. $e - \int\limits_0^1 {{e^x}dx} $
B. $e - \int\limits_1^e {\ln ydy} $
C. $e - 1$
D. $1$

Answer 1

Hint: To solve this question, we need to know the basic theory related to the area under a curve. As we know the area under a curve that exists between two points can be calculated by conducting a definite integral between the two points. Here we have a curve $y = {e^x}$. First, we will shade the region which consists of Area bounded by $y = {e^x}$ and lines x=0 and y=e as shown below and then after apply definite integral between the two points.

Complete step-by-step answer:
As we know, the area under the curve y = f(x) between x = a & x = b, one must integrate y = f(x) between the limits of a and b.

In the above figure, Area bounded by $y = {e^x}$ and lines x=0 and y=e is represented as a shaded region of the graph.
From figure we conclude that-
In the graph, the upper curve will be y = e and the lower one will be $y = {e^x}$.
Required area is,
A = $\int\limits_1^e {\ln ydy} $
= $e - \int\limits_0^1 {{e^x}dx} $
= $e - \left( {{e^x}} \right)_0^1$
= $e - \left( {e - 1} \right)$
= e – e + 1
= 1
Therefore, option (A) and (D) are the correct answer.

Note: Always remember that the first and the most important step is to plot the two curves on the same graph. If one can’t plot the exact curve, at least an idea of the relative orientations of the curves should be known.