Question

Math

Posted 3 months ago

```
Which integral gives the length of the graph of $f(x)=\ln \left(x^{2}\right)$ between $x=a$ and $x=b ?$
Choose 1 answer:
(A) $\int_{a}^{b} \sqrt{1+2 \ln x} d x$
(B) $\int_{a}^{b} \sqrt{1+(2 \ln x)^{2}} d x$
(c) $\int_{a}^{b} \sqrt{1+\frac{2}{x}} d x$
(D) $\int_{a}^{b} \sqrt{1+\frac{4}{x^{2}}} d x$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 2

First, we need to find the derivative of $f(x) = \ln(x^2)$. Using the chain rule, $f'(x) = \frac{1}{x^2} \cdot 2x = \frac{2}{x}$

step 3

Now, we square the derivative: $\left(\frac{2}{x}\right)^2 = \frac{4}{x^2}$

step 4

Substitute the squared derivative into the arc length formula: $L = \int_{a}^{b} \sqrt{1 + \frac{4}{x^2}} \, dx$

step 5

This matches option (D) from the multiple choice answers provided

D

Key Concept

Arc Length of a Curve

Explanation

The arc length of a curve defined by a function $f(x)$ from $x=a$ to $x=b$ is given by the integral $\int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx$. To find this, we need to compute the derivative of the function, square it, and then integrate.

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