Question

Fill in the blank: The length of the graph of $y=x^2$on $[2,4]$is equal to the area under the graph of the function ________ on the same interval.

Short answer

The length of the graph of $y=x^2$on [2, 4] is equal to the area under the graph of the function $\sqrt{1+4x^2}$on the same interval.

Step-by-step solution

Given

The given function is$y=x^2$on the interval$\left[2,4\right]$.

To find

Fill in the blanks.

Calculation

Using derivative we get:

$$\begin{gathered} y=x^2 \\ \frac{dy}{dx}=2x \end{gathered}$$

So the length of the arc is:

$$\begin{gathered} {\int }_2^4\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx \\ ={\int }_2^4\sqrt{1+{\left(2x\right)}^2}dx \\ ={\int }_2^4\sqrt{1+4x^2}dx \end{gathered}$$

Hence, the length of the graph of $y=x^2$on [2, 4] is equal to the area under the graph of the function$\sqrt{1+4x^2}$on the same interval.