Q. 6 - Calculus

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.

Step-by-Step Solution

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 1: Given

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

Step 2: To find

Fill in the blanks.

Step 3: 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.