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) $\int_{a}^{b} \sqrt{1+\frac{4}{x^{2}}} d x$

Question

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) $\int_{a}^{b} \sqrt{1+\frac{4}{x^{2}}} d x$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the length of the graph of a function $f(x)$ from $x=a$ to $x=b$, we use the arc length formula: $L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx$. ‖ ‖ 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.⚹◊What is the formula for calculating the length of a curve defined by a given function between two points?⍭ Generate me a similar question◊

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