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Problem: Convert the double integral over the area \( G \) to the iterated integral in Cartesian and polar coordinates: \[ \iint_{(G)} f(x, y) d x d y \] \( (G) \) \[ G=\left\{(x, y) \mid 0 \leq y \leq 1, \frac{y^{2}}{3} \leq x \leq \sqrt{3-y^{2}}\right\} . \]

9.1.9 Double integrals

2.55 $

Problem: Calculate the double integral of the function \( x y \) over the area \( G \), given by the crossing of lines given parametrically: \[ G:\left\{\begin{array}{l} x=a \cos ^{3} t \\ y=a \sin ^{3} t \\ x=0, y=0 \end{array}\left(0 \leq t \leq \frac{\pi}{2}\right) \quad \iint_{G} x y d x d y-?\right. \]

9.1.10 Double integrals

2.55 $

Problem: Find the limit: \[ \lim _{x \rightarrow 2} \frac{\log (5-2 x)}{\sqrt{10-3 x}-2} . \]

2.1.19 Calculation of limits

0.51 $

Problem: Find the limit: \[ \lim _{x \rightarrow 1} \frac{\cos \frac{\pi x}{2}}{5-5^{x}} \]

2.1.20 Calculation of limits

0.51 $

Problem: Give the definition of the one-sided limit in the language \( \varepsilon-\delta \).

2.1.21 Calculation of limits

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Problem: Find the limit: \[ \lim _{x \rightarrow-1} \frac{x^{3}-3 x-2}{x^{3}-2 x^{2}-x-2} \text {. } \]

2.1.22 Calculation of limits

0.51 $

Problem: Find the limit: \[ \lim _{x \rightarrow \infty}\left(\sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-1)^{2}}\right) \]

2.1.23 Calculation of limits

1.02 $

Problem: Find the limit: \[ \lim _{x \rightarrow \infty} \frac{\sin 7 \pi x}{\tan 8 \pi x} . \]

2.1.24 Calculation of limits

0.77 $

Problem: Find the limit: \[ \lim _{x \rightarrow \alpha} \frac{\alpha^{x^{2}-\alpha^{2}}-1}{\tan \ln \frac{x}{\alpha}} \text {. } \]

2.1.25 Calculation of limits

1.02 $

Problem: Find the limit: \[ \lim _{x \rightarrow 0} \sqrt[x^{2}]{2-\cos x} \text {. } \]

2.1.26 Calculation of limits

1.28 $

Problem: Compare the infinitesimals: \[ \alpha(x)=\ln \left(x^{2}+1\right), \quad \beta(x)=x^{3}-2 x, x \rightarrow 0 . \]

2.12.8 Asymptotic analysis

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Problem: Compare the infinitesimals: \[ \alpha(x)=\sin \ln (x+1), \beta(x)=x^{2}+2 x, x \rightarrow 0 . \]

2.12.9 Asymptotic analysis

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Problem: Compare the infinitesimals: \[ \begin{array}{l} \alpha(x)=\sqrt{\cos \ln x}-1, \\ \beta(x)=x^{3}-3 x+2, \quad x \rightarrow 1 . \end{array} \]

2.12.10 Asymptotic analysis

1.28 $

Problem: Select the main part of the form \( c x^{k} \) of the infinitesimal function: \[ \alpha(x)=\cos x-\cos 5 x, \quad x \rightarrow 0 . \]

2.12.14 Asymptotic analysis

1.02 $

Problem: Select the main part of the form \( c x^{k} \) of the infinitesimal function: \[ \alpha(x)=\sin \ln \left(x^{3}+1\right), \quad x \rightarrow 0 . \]

2.12.15 Asymptotic analysis

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