Question
Show that if the arc length of $f(x)$ over $[0, a]$ is proportional to $a$then $f(x)$ must be a linear function.
Step 1
Step 1: The arc length of a function $f(x)$ over the interval $[0, a]$ is given by the integral \[ \int_0^a \sqrt{1 + (f'(x))^2} \, dx \] where $f'(x)$ is the derivative of $f(x)$. Show more…
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Show that if the arc length of $y=f(x)$ over $[0, a]$ is proportional to $a,$ then $y=f(x)$ must be a linear function.
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Let $a, r>0 .$ Show that the arc length of the curve $x^{r}+y^{r}=a^{r}$ for $0 \leq x \leq a$ is proportional to $a$ .
FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS
Prove, in two ways, that the arc length of a linear function $f(x)=m x+c$ on an interval $[a, b]$ is equal to $(b-a) \sqrt{1+m^{2}}$ (a) by using the distance formula; (b) by using Theorem 6.7 .
Applications of Integration
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