Question

Area bounded by the curves $y = {e^x}$, $y = {e^{ - x}}$ and the straight line $x = 1$ is (in sq. units)
A. $e + \dfrac{1}{e}$
B. $e + \dfrac{1}{e} + 2$
C. $e + \dfrac{1}{e} - 2$
D. $e - \dfrac{1}{e} + 2$
E. $e - \dfrac{1}{e}$

Answer 1

Hint: In this question, we are given the equation of the curves $y = {e^x}$, $y = {e^{ - x}}$ and the line i.e., $x = 1$. We have to find the area bounded by them. Firsly, plot the graph of the given equations of curve and the line. Then, see the point of intersection of both the curves. To calculate the area, integrate the difference of first curve and the second curve with respect to $dx$ from $0$ to $1$. Solve it further using the integration formula of the exponential function.



Formula Used:Integration of the exponential function –
$\int {{e^x}dx = {e^x}} $



Complete step by step solution:Given that,
Equation of the curves $y = {e^x}$, $y = {e^{ - x}}$ and the straight line $x = 1$
Graph of the given equations is attached below (figure 1);


Figure 1: A graph contains the plot of given equation of the curves and the line
Now, you can see both the curves meet at $x = 0$.
So, we’ll calculate the area of the bounded region by integrating the difference of first curve and the second curve with respect to $dx$ from the line $x = 0$ to $x = 1$
Therefore, the area of the bounded region will be;
$A = \int\limits_0^1 {\left( {{e^x} - {e^{ - x}}} \right)} dx$
Now, as we know that the integration of the function ${e^x}$ is ${e^x}$
On integrating the function, we get $A = \left[ {{e^x} + {e^{ - x}}} \right]_0^1$
On resolving the limits, we get $A = {e^1} + {e^{ - 1}} - {e^0} - {e^{ - 0}}$
$A = e + \dfrac{1}{e} - 2$ sq. units



Option ‘C’ is correct

Note: Different methods are used to determine the area under the curve, with the antiderivative method being the most prevalent. The area under the curve can be calculated by knowing the curve's equation, borders, and the axis surrounding the curve. There exist formulas for obtaining the areas of regular shapes such as squares, rectangles, quadrilaterals, polygons, and circles, but no formula for finding the area under a curve. The integration procedure aids in solving the equation and determining the required area. Antiderivative methods are highly useful for determining the areas of irregular planar surfaces.