Q. 7 - Calculus

Question

Fill in the blank: The area under the graph of $y=\sqrt{1+e^{2x}}$on [0, 2] is equal to the length of the graph of the function ____ on the same interval.

Step-by-Step Solution

Answer

The area under the graph of $y=\sqrt{1+e^{2x}}$ on [0, 2] is equal to the length of the graph of the function $y=e^x$ on the same interval.

Step 1: Given

The area under the graph of $y=\sqrt{1+e^{2x}}$ on [0, 2] is equal to the length of the graph of the function ____ on the same interval.

Step 2: To find

Fill in the blank.

Step 3: Calculation

We compare $\sqrt{1+e^{2x}}$ with $\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$.

Comparing we get: 

$$\begin{gathered} \frac{dy}{dx}=e^x \\ y=\int e^{x}dx \\ y=e^x \end{gathered}$$

Hence, the area under the graph of $y=\sqrt{1+e^{2x}}$ on [0, 2] is equal to the length of the graph of the function $y=e^x$ on the same interval.