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Problem: Calculate the double integral. Change the order of integration without calculating it. \[ \iint_{D}(x+y) d x d y \text {, where } D: y \geq x, y \leq 2 x, x+y \leq 6 \text {. } \]

9.1.35 Double integrals

2.06 $

~~2.57 $~~

Problem: Find out which surface is defined by the following equation and schematically depict it: \[ x^{2}+z^{2}-4 x-4 z+4=0 . \]

3.5.2 Surfaces of the 2-nd order

0.77 $

Problem: Find out which surface is defined by the following equation and schematically depict it: \[ x^{2}+2 y^{2}+z^{2}-2 x y-2 y z+4=0 \text {. } \]

3.5.3 Surfaces of the 2-nd order

0.77 $

Problem: Let the given surface be $ \sigma: 9(x-1)^{2}+4 y^{2}-36 z^{2}=36 $ and the points be $ M_{1}(1 ; 0 ; 0), M_{2}(3 ; 1 ; 1) $. 1) Define the type of the surface $ \sigma $, 2) Draw the surface $ \sigma $, 3) Find sections of the surface $ \sigma $ with coordinate planes and define their foci and asymptotes, 4) Define the location of the points $ M_{1} $ and $ M_{2} $ in relation to the surface $ \sigma $ 5) Find the points of intersection of the straight line passing through points $ M_{1} $ and $ M_{2} $ with the surface $ \sigma $.

3.5.4 Surfaces of the 2-nd order

5.14 $

35 Problem: Calculate the area of the figure bounded by the cardioid $ r=3(1+\cos \varphi) $.

9.7.6 The area of a region

1.28 $

Problem: Applying differentiation with respect to the parameter, calculate the following integral: \[ I(a)=\int_{0}^{\frac{\pi}{2}} \frac{\tan ^{-1}(a \sin x)}{\sin x} d x . \]

9.2.4 Integrals depending on a parameter

3.85 $

Problem: Applying differentiation with respect to the parameter, calculate the following integral: \[ I(\alpha, \beta)=\int_{0}^{+\infty} \frac{\ln \left(1+\alpha^{2} x^{2}\right) \tan ^{-1}(\beta x)}{x^{3}} d x, \quad \alpha, \beta>0 . \]

9.2.5 Integrals depending on a parameter

5.14 $

Problem: Prove that the integral $ \mathrm{I}(\alpha) $ converges uniformly on the set $ \mathrm{E} $, if: \[ I(a)=\int_{2}^{+\infty} x^{\alpha} e^{-2 x^{\alpha}} d x, \quad \mathrm{E}=[1 ; 2] . \]

9.2.6 Integrals depending on a parameter

2.57 $

Problem: Using the Dirichlet/Frullani/Fresnel/Euler/Poisson/Laplace integrals, calculate the following integral: \[ I(\alpha)=\int_{0}^{+\infty} e^{-\left(x^{2}+\frac{\alpha^{2}}{x^{2}}\right)} d x, \quad \alpha>0 . \]

9.2.7 Integrals depending on a parameter

3.85 $

Problem: Using the Dirichlet/Frullani/Fresnel/Euler/Poisson/Laplace integrals, calculate the following integral: \[ I=\int_{0}^{+\infty} \frac{\sin ^{5} \alpha x}{x^{3}} d x . \]

9.2.8 Integrals depending on a parameter

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Problem: Using the Euler integral, calculate the integral: \[ I=\int_{0}^{+\infty} \frac{x^{\alpha} \ln ^{2} x}{1+x^{2}} d x, \quad|\alpha|<1 . \]

9.2.9 Integrals depending on a parameter

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Problem: Investigate the continuity of the function $ F(y) $ on the set $ Y $ : \[ F(y)=\int_{0}^{+\infty} y \cdot e^{-y^{2} x} d x, \quad Y=\mathbb{R} . \]

9.2.10 Integrals depending on a parameter

2.57 $

Problem: Find the coordinates of the center and the radius of the circle: \[ x^{2}+y^{2}-8 x+6 y=0 \text {. } \] 밤 $ (\mathrm{Ctrl}) \cdot $

3.1.5 Curves of the 2-nd order

0 $

Problem: Write the equation of a straight line passing through the left focus and the lower vertex of the ellipse \[ \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 . \]

3.1.6 Curves of the 2-nd order

1.03 $

Problem: Find the equation of a hyperbola the vertices and foci of which are at the corresponding foci and vertices of the ellipse \[ \frac{x^{2}}{8}+\frac{y^{2}}{5}=1 \text {. } \]

3.1.7 Curves of the 2-nd order

1.28 $