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3) Problem: Calculate the volume of the solid, formed by rotating around the axis \( O x \) of the figure, bounded by the lines: \[ 2 x-y-2=0, y=0, x=3 \] Let's make a drawing.

9.6.4 Volume of a solid of revolution

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(3) Problem: Show that the vector field \( \vec{a} \) is potential and find its potential: \[ \begin{array}{l} \vec{a}=y z \cos x y \cdot \vec{\imath}+z x \cos x y \cdot \vec{\jmath}+\sin x y \cdot \vec{k}, \\ M_{1}(1,1,1), \quad M_{2}(0,0,0) \end{array} \]

9.9.1 Potential and solenoidal fields

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T Problem: Calculate the volume of the solid formed by rotating around the axis \( O x \) of the figure, bounded by the lines: \( y=x^{2}, y=1 \). Let's plot the drawing.

9.6.3 Volume of a solid of revolution

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Problem: Calculate the integral of a function of a complex variable along the given curve. \[ \int_{L} z \operatorname{Re} z^{2} d z, L:\{|z|=1, \operatorname{Im} z \geq 0\} \text {. } \]

10.1.1 Integral of a complex variable

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Problem: Calculate the integral. \[ \oint_{|z+3 i|=2} \frac{4}{(z-1+3 i)^{2}(z-3+3 i)} d z \]

10.1.2 Integral of a complex variable

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Problem: Calculate the integral. \[ \oint_{C} \frac{\sin z d z}{\left(z^{2}-\frac{\pi^{2}}{4}\right)^{2}}, \] \( C:\left|z-\frac{\pi}{2}\right|=1 \).

10.1.3 Integral of a complex variable

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Problem: Find the singular points of the function, examine their type and calculate their residues: \[ f(z)=z^{5} \sin \frac{1}{z^{2}} \text {. } \]

10.2.1 Singular points and residues

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Problem: Complex numbers \( z_{1}, z_{2} \) are given. Find \[ z_{1}+z_{2}, z_{1}-z_{2}, z_{1} \cdot z_{2}, \frac{z_{1}}{z_{2}}, \] where \( z_{1}=2-3 i, z_{2}=5-4 i \).

10.3.1 Operations with complex numbers

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Problem: Find all the values of the root \[ \sqrt[4]{-\frac{1}{16}} . \]

10.3.2 Operations with complex numbers

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Problem: Draw a domain, given by the inequalities. \[ |z| \leq 1, \quad \arg (z+i)>\frac{\pi}{4} . \]

10.3.3 Operations with complex numbers

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Problem: Find the Laplace transformation of the periodic function \( f(t)=B|\sin \omega t| \).

10.4.1 Laplace transform

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Problem: Integrate the following linear differential equation under the given initial conditions: \[ x^{\prime \prime}+x^{\prime}-2 x=e^{t}, \quad x(0)=x_{0}, x^{\prime}(0)=x_{0}{ }^{\prime} . \] clip2net.com

8.2.1.1 Differential equations

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Problem: Solve the Cauchy problem using the operational method. \[ 2 y^{\prime \prime}+5 y^{\prime}=29 \cos t, y(0)=-1, y^{\prime}(0)=0 . \] clip2net.com

8.2.1.3 Differential equations

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Problem: Solve the system of differential equations, using the operational method. \[ \left\{\begin{array}{l} \dot{x}=2 x+8 y \\ \dot{y}=3 x+4 y, \quad x(0)=0, \quad y(0)=2 . \end{array}\right. \]

8.2.2.1 Systems of differential equations

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Problem: Find the solutions \( y=y(x) \) of the differential equation, that are non-identically zero in the indicated area and satisfy the given boundary conditions (the Sturm-Liouville problem). \[ \left\{\begin{array}{c} y^{\prime \prime}+\lambda y=0, \quad \frac{3}{4} \leq x \leq 1 \\ y\left(\frac{3}{4}\right)=y^{\prime}(1)=0 \end{array} \quad y \not \equiv 0 .\right. \]

11.1.1 Sturm-Liouville problem

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