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3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} z=6 \sqrt{x^{2}+y^{2}} \\ z=16-x^{2}-y^{2} \end{array} .\right. \]

9.5.4 Volume of a solid

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Problem: A circle with a radius of \( 1 \mathrm{~cm} \) is randomly thrown on the plane, ruled by parallel lines, at the distance of 6 from each other. Find the probability that the circle doesn't cross any of the lines.

15.6.5 Definition and properties of probability

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Problem: Applying differentiation with respect to the parameter, calculate the following integral: \[ I(a)=\int_{0}^{\frac{\pi}{2}} \ln \left(a^{2}-\cos ^{2} \varphi\right) d \varphi,|a|>1 . \]

9.2.11 Integrals depending on a parameter

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Problem: Applying differentiation with respect to the parameter, calculate the following integral: \[ \begin{array}{l} I=\int_{0}^{+\infty} \frac{e^{-\alpha x}-e^{-\beta x}}{x} \sin m x d x, \\ \alpha>0, \quad \beta>0, \quad m \neq 0 . \end{array} \]

9.2.12 Integrals depending on a parameter

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients. \[ y^{I V}+2 y^{\prime \prime}-2 y=0 \text {. } \] clip2net.com

8.1.3.16 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients. \[ 25 y^{I V}+10 y^{2}+y=0 \text {. } \] clip2net.com

8.1.3.17 Higher order differential equations

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Problem: Find the general solution of the homogeneous differential equation with constant coefficients: \[ y^{\prime \prime}-5 y^{\prime}+6 y=3 x^{2}-5 x-5 \text {. } \] clip2net.com

8.1.2.18 Second order differential equations

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Problem: Find the general solution of the linear inhomogeneous differential equation with constant coefficients. \[ y^{\prime \prime}+2 y^{\prime}-3 y=5 e^{2 x} \text {. } \] clip2net.com

8.1.2.19 Second order differential equations

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Problem: Find the general integral of the differential equation. (Present the answer in the form of \( \varphi(x, y)=C) \). \[ (x+1) y^{\prime}+y(y+1)=0 . \]

8.1.1.20 First order differential equations

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Problem: Find the general integral of the differential equation. \[ x y^{\prime}=\frac{y^{2}}{x}+y+x \text {. } \]

8.1.1.21 First order differential equations

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Problem: Find the general solution of the linear inhomogeneous differential equation with constant coefficients. \[ x y^{\prime}=y-2 x^{2} \text {. } \]

8.1.1.22 First order differential equations

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Problem: Find the general integral of the differential equation: \[ \left(2 x-1-\frac{y}{x^{2}}\right) d x-\left(2 y-\frac{1}{x}\right) d y=0 . \]

8.1.1.23 First order differential equations

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Problem: Solve the Cauchy problem: \[ \left\{\begin{array}{l} y^{\prime}-\frac{y}{x}=x y^{2} \\ y(1)=\frac{2}{3} \end{array}\right. \]

8.1.1.24 First order differential equations

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Problem: Find the general solution of the third order linear inhomogeneous differential equation: \[ y^{\prime \prime \prime} \tan 5 x=5 y^{\prime \prime} \text {. } \]

8.1.3.25 Higher order differential equations

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Problem: The one-dimensional random variable \( \xi \) is given by the distribution density \( P(x)=\gamma e^{a x^{2}+b x+c} \), where \( a=-3, b=-4, c=0, x_{1}=\frac{1}{3}, \quad x_{2}=\frac{4}{3} \). Find the constant \( \gamma \), the expected value, the distribution function and the probability that the value \( \xi \) belongs to the integral \( \left[x_{1}, x_{2}\right] \).

15.2.7 One dimensional random variables and their characteristics

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