Example 2.46. Find the average value of the function $u(x)=$ $=1 / \sqrt{x}$ on the half-interval $x \in(0,1]$.
Solution.

$$
M(x)=\lim _{a \rightarrow+0}\left(\int_{a}^{1} \frac{d x / \sqrt{x}}{1-0}\right)=\lim _{a \rightarrow+0} \frac{2 \sqrt{1}-2 \sqrt{a}}{1}=2
$$

## 2.4. GEOMETRIC APPLICATIONS OF DEFINITE INTEGRALS

## Area of a Plane Curve

The area of a plane figure bounded by curves given by their equations in Cartesian coordinates: above - $y=y_{1}(x)$, below - $y=y_{2}(x)$, and on the left and right by segments of the lines $x=a$ and $x=b$, is calculated by the formula

$$
S=\left|\int_{a}^{b}\left(y_{1}(x)-y_{2}(x)\right) d x\right|
$$

The area of a plane figure bounded by curves given by their equations in Cartesian coordinates: to the right - $x=x_{1}(y)$, to the left - $x=x_{2}(y)$, and below and above by segments of the lines $y=c$ and $y=d$, is calculated by the formula

$$
S=\left|\int_{c}^{d}\left(x_{1}(y)-x_{2}(y)\right) d y\right|
$$

The area of a closed plane figure bounded by a curve given by parametric equations: $x=x(t)$, $y=y(t), 0 \leqslant t \leqslant T$ can be calculated using one of three formulas (assuming that as $t$ increases from 0 to $T$, the curve is traversed counterclockwise).

$$
S=\int_{0}^{T} y(t) x^{\prime}(t) d t=\int_{0}^{T} x(t) y^{\prime}(t) d t=0.5 \int_{0}^{T}\left(x(t) y^{\prime}(t)+y(t) x^{\prime}(t)\right) d t
$$

The area of a figure bounded by a curve given by an equation in polar coordinates $\rho=\rho(\varphi)$ and segments of two rays $\varphi=\alpha$ and $\varphi=\beta$, is calculated by the formula

$$
S=0.5\left|\int_{\alpha}^{\beta}(\rho(\varphi))^{2} d \varphi\right|
$$

## Arc Length

The length of an arc of a smooth curve given by the equation in Cartesian coordinates $y=y(x)(a \leqslant x \leqslant b)$, is calculated by the formula

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

The length of an arc of a smooth curve given by the equation in Cartesian coordinates $x=x(y)(c \leqslant y \leqslant d)$, is calculated by the formula

$$
L=\left|\int_{c}^{d} \sqrt{1+\left(x^{\prime}(y)\right)^{2}} d y\right|
$$

The length of an arc of a smooth curve given by parametric equations $x=x(t)$ and $y=y(t)\left(t_{1} \leqslant t \leqslant t_{2}\right)$ is calculated by the formula

$$
L=\left|\int_{t_{1}}^{t_{2}} \sqrt{\left(x^{\prime}(t)\right)^{2}+\left(y^{\prime}(t)\right)^{2}} d t\right|
$$

The length of an arc of a smooth curve given by parametric equations in three-dimensional space $x=x(t), y=y(t)$, $z=z(t)\left(t_{1} \leqslant t \leqslant t_{2}\right)$ is calculated by the formula

$$
L=\left|\int_{t_{1}}^{t_{2}} \sqrt{\left(x^{\prime}(t)\right)^{2}+\left(y^{\prime}(t)\right)^{2}+\left(z^{\prime}(t)\right)^{2}} d t\right|
$$

The length of an arc of a smooth curve given by a polar equation, $\rho=\rho(\varphi), \alpha \leqslant \varphi \leqslant \beta$ is calculated by the formula

$$
L=\left|\int_{a}^{\beta} \sqrt{\left.\left.(\rho(\varphi))^{2}\right)+\left(\rho^{\prime}(\varphi)\right)^{2}\right)} d \varphi\right|
$$

## Surface Area of Revolution

If a segment of a smooth curve is rotated around an arbitrary axis, the area of the surface formed is calculated by the formula

$$
S=2 \pi\left|\int_{r_{1}}^{r_{2}} G(l(r)) d l(r)\right|
$$

where $d l$ is the differential of the arc $l, G(l(r))$ is the distance from a point on the curve to the axis of rotation, $r_{1}$ and $r_{2}$ are the values of the arc variable $r$ corresponding to the ends of the arc.

If a segment of a smooth curve given by the equation in Cartesian coordinates $y=f(x)(a \leqslant x \leqslant b)$ is rotated around the $O X$ axis, the area of the surface formed is calculated by the formula

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

If a segment of a smooth curve given by the equation in Cartesian coordinates $x=g(y)(c \leqslant x \leqslant d)$ is rotated around the $O Y$ axis, the area of the surface formed is calculated by the formula

$$
S_{y}=2 \pi\left|\int_{a}^{b} g(y) \sqrt{1+\left(g^{\prime}(y)\right)^{2}} d y\right|
$$

If a segment of a smooth curve given by parametric equations $x=x(t)$ and $y=y(t)\left(t_{1} \leqslant t \leqslant t_{2}\right)$ is rotated around the $O X$ axis, the area of the surface formed is calculated by the formula

$$
S_{x}=2 \pi\left|\int_{t_{1}}^{t_{2}} y(t) \sqrt{\left(x^{\prime}(t)\right)^{2}+\left(y^{\prime}(t)\right)^{2}} d t\right|
$$

If a segment of a smooth curve given by parametric equations $x=x(t)$ and $y=y(t)\left(t_{1} \leqslant t \leqslant t_{2}\right)$ is rotated around the $O Y$ axis, the area of the surface formed is calculated by the formula

$$
S_{y}=2 \pi\left|\int_{t_{1}}^{t_{2}} x(t) \sqrt{\left(x^{\prime}(t)\right)^{2}+\left(y^{\prime}(t)\right)^{2}} d t\right|
$$

If a segment of an arc given by the equation in polar coordinates $\rho=\rho(\varphi)(\alpha \leqslant \varphi \leqslant \beta)$ is rotated around the polar axis, the area of the surface of revolution is calculated by the formula

$$
S_{x}=2 \pi\left|\int_{\alpha}^{\beta} \rho(\varphi) \sin \varphi \sqrt{(\rho(\varphi))^{2}+\left(\rho^{\prime}(\varphi)\right)^{2}} d \varphi\right|
$$

Volume of a Solid

Let $S(x)$ be a continuous function on the interval $[a, b]$ which is equal to the area of a cross-section perpendicular to the $O X$ axis of some solid, then the volume of this solid is calculated by the formula

$$
V=\left|\int_{a}^{b} S(x) d x\right|
$$

Let $Q(y)$ be a continuous function on the interval $[c, d]$ which is equal to the area of a cross-section perpendicular to the $O Y$ axis of some solid, then the volume of this solid is calculated by the formula

$$
V=\left|\int_{a}^{b} Q(y) d y\right|
$$

If the curvilinear trapezoid $y=y(x) \geqslant 0, a \leqslant x \leqslant b$ is rotated around the $O X$ axis, the volume of the resulting solid of revolution is found by the formula

$$
V=\pi \int_{a}^{b}(f(x))^{2} d x
$$

If the curve is given parametrically $x=x(t) \geqslant 0, y=$ $=y(t) \geqslant 0, t_{1} \leqslant t \leqslant t_{2}$, and on this segment $x^{\prime}(t) \geqslant 0$, then the volume of the solid of revolution around the $O X$ axis is

$$
V=\left|\pi \