## Problem Statement

Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.

$$
y=\operatorname{ch} x+3, \quad 0 \leq x \leq 1
$$
## Solution

The length of the arc of a curve defined by the equation $y=f(x) ; a \leq x \leq b$ is given by the formula

$$
L=\int_{a}^{b} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} d x
$$

Let's find the derivative of the given function:

$$
f^{\prime}(x)=(\operatorname{ch} x+3)^{\prime}=\operatorname{sh} x
$$

Then, using the formula above, we get:

$$
\begin{aligned}
L & =\int_{0}^{1} \sqrt{1+(\operatorname{sh} x)^{2}} d x= \\
& =\int_{0}^{1} \operatorname{ch} x d x=\left.\operatorname{sh} x\right|_{0} ^{1}=\operatorname{sh} 1-\operatorname{sh} 0=\operatorname{sh} 1
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+17-23$ »

Categories: Kuznetsov Problem Book Integrals Problem 17 | Integrals | Popular Problems

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## Problem Kuznetsov Integrals 17-24

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