Double integrals

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Problem: Change the order of integration: \[ I=\int_{0}^{a} d y \int_{\frac{y^{2}}{2 a}}^{a-\sqrt{a^{2}-y^{2}}} f d x+\int_{0}^{a} d y \int_{a+\sqrt{a^{2}-y^{2}}}^{2 a} f d x+\int_{0}^{2 a} d y \int_{\frac{y^{2}}{2 a}}^{2 a} f d x . \]

9.1.31 Double integrals

1.2 $

Problem: Find the mass of the flat figure \( D \) with density \( \rho \) : \[ \rho(x, y)=|y|, \quad D:(x-3)^{2}+y^{2} \leq 1, \quad x \geq 3 . \]

9.1.32 Double integrals

1.2 $

Problem: Calculate the area of the region: \[ D: y=20-x^{2}, y=-8 x \text {. } \]

9.1.33 Double integrals

0.96 $

Problem: Make a replacement of variables in the double integral: \[ G:\left\{\begin{array}{l} y=x \\ x^{2}+y^{2}=9 \\ x y=4(x>0, y \geq x) \end{array},\left\{\begin{array}{l} u=x^{2}+y^{2}-9 \\ v=x y-4 \end{array} .\right.\right. \]

9.1.11 Double integrals

1.45 $

Problem: Calculate the value of the iterated integral: \[ I=\int_{0}^{1} d x \int_{0}^{1} \frac{x d y}{\frac{\left(1+x^{2}+y^{2}\right)^{3}}{2}} \]

9.1.34 Double integrals

1.45 $

Problem: Calculate the coordinates of the center of gravity of the flat homogeneous plate \( D \) with density \( \rho \) : \[ D:\left\{\begin{array}{l} x=y^{2}-3 \quad \rho=1 . \\ y=x \end{array}\right. \]

9.1.36 Double integrals

2.89 $

Problem: Determine the sign of the value of the double integral over the region \( G \) : a) \( \iint_{|x|+|y| \leq 1} \ln \left(x^{2}+y^{2}\right) d x d y, \quad G:\{|x|+|y| \leq 1\} \), b) \( \iint_{1 \leq x^{2}+y^{2} \leq 4} \ln \left(x^{2}+y^{2}\right) d x d y, \quad G:\left\{1 \leq x^{2}+y^{2} \leq 4\right\} \).

9.1.37 Double integrals

2.41 $

Problem: In the double integral, make the transition to the iterated integral: \[ \iint_{G} f(x) d x d y \text {, } \] a) \( G:\left\{\begin{array}{c}x=1 \\ y=x^{2}, y=2 x(x \leq 1)\end{array}\right. \), b) \( G:\left\{\begin{array}{c}x=0, y=0 \\ x^{2}+y^{2}=a^{2}(x \geq 0)\end{array} \quad(a \geq 0)\right. \).

9.1.38 Double integrals

2.41 $

Problem: Estimate the double integral from above and below: \[ I=\iint_{G}(4+\cos x y) d x d y, G:\left\{(x, y) \mid x^{2}+y^{2} \leq 4\right\} \text {. } \]

9.1.39 Double integrals

1.2 $

Problem: Make a change of variables in iterated integrals: \[ \begin{array}{l} \text { a) } I=\int_{a}^{b} d x \int_{\alpha x}^{\beta x} f(x, y) d y, \\ 0

9.1.40 Double integrals

3.61 $

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