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.

Short 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-by-step solution

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.

To find

Fill in the blank.

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.