Question

Обчислити об,єму тіл обертання f(x)=2/x x1=2 x2=5 пожалуста очень надо сегодня

Answer 1

Дано: f(x)=2/x, a=2, b=5
 найти: V

решение.
$V= \int\limits^a_b { \pi * f^{2} (x)} \, dx$

$\int\limits^5_2 { \pi *( \frac{2}{x} ) ^{2} } \, dx = \pi * \int\limits^5_2 { \frac{4}{ x^{2} } } \, dx =4 \pi * \frac{ x^{-2+1} }{-2+1} | _{2} ^{5} = \frac{4 \pi }{-1}* x^{-1}| _{2} ^{5} =$
$=- \frac{4 \pi }{x} | _{2} ^{5} =-4 \pi *( \frac{1}{5}- \frac{1}{2} )=-4 \pi *(- \frac{3}{10} )=1,2 \pi$см³

Answer 2

Calculating the Volume of Revolution

To calculate the volume of revolution of the function \( f(x) = \frac{2}{x} + x \) from \( x_1 = 2 \) to \( x_2 = 5 \), we can use the method of cylindrical shells.

Method of Cylindrical Shells

The volume \( V \) of the solid formed by revolving the curve \( y = f(x) \) around the x-axis from \( x = a \) to \( x = b \) can be calculated using the formula:

\[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \]

Where \( f(x) \) is the function defining the curve, and \( a \) and \( b \) are the limits of integration.

Applying the Method to the Given Function

Using the given function \( f(x) = \frac{2}{x} + x \) and the limits \( x_1 = 2 \) and \( x_2 = 5 \), we can calculate the volume of revolution using the formula:

\[ V = 2\pi \int_{2}^{5} x \cdot \left(\frac{2}{x} + x\right) \, dx \]

Solving this integral will give us the volume of the solid formed by revolving the curve around the x-axis within the given limits.

Let's calculate the integral and find the volume.