Double integrals

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Problem: Change the order of integration in iterated integral: \[ \int_{-2}^{-1} d x \int_{-(x+2)}^{0} f d y+\int_{-1}^{0} d y x \int_{\sqrt[3]{x}}^{0} f d y . \]

9.1.1 Double integrals

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Problem: Calculate the values of double integrals: 1) $ \iint_{(D)}\left(8 x y+18 x^{2} y^{2}\right) d x d y, \quad D:\left\{\begin{array}{c}x=1 \\ y=\sqrt[3]{x} \\ y=-x^{2}, \quad(x \geq 0)\end{array}\right. $ 2) $ \iint 4 y^{2} \sin 2 x y d x d y, D:\left\{\begin{array}{c}x=0 \\ y=\sqrt{2 \pi} \\ y=2 x\end{array}\right. $

9.1.2 Double integrals

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Problem: Calculate the mass of the plate $ D $ with density $ \rho $ : \[ D:\left\{\begin{array}{l} x^{2}+y^{2}=4 \\ x^{2}+y^{2}=16 \end{array}, \quad \rho=\frac{2}{x^{2}+y^{2}}\right. \]

9.1.3 Double integrals

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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

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Problem: Calculate the mass of the plate $ D $ with density, where \[ \mu=\frac{2 y-3 x}{x^{2}+y^{2}}, \quad D:\left\{\begin{array}{l} x^{2}+y^{2}=4 \\ x^{2}+y^{2}=16 \\ x=0, y=0,(x \leq 0, y \geq 0) \end{array}\right. \]

9.1.4 Double integrals

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Problem: Calculate the mass of the plate $ D $ with the density, where \[ D:\left\{\begin{array}{l} \frac{x^{2}}{16}+y^{2} \leq 1 \\ x \geq 0, y \geq 0 \end{array} \quad \mu(x, y)=5 x y^{7} .\right. \]

9.1.5 Double integrals

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Problem: Find the double integral over the area \[ D:\left\{\begin{array}{l} x=1, y=x^{3} \\ y=-\sqrt[3]{x} \end{array}\right. \text {. } \]

9.1.6 Double integrals

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Problem: Change the order of integration in the iterated integral: \[ \int_{0}^{1} d x \int_{0}^{x} f d y+\int_{1}^{\sqrt{2}} d x \int_{0}^{\sqrt{2-x^{2}}} f d y . \]

9.1.7 Double integrals

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Problem: Find the statistical moment about the axis $ O X $ of the plate $ D $ with density, where \[ D:\left\{\begin{array}{c} x=\sqrt{4-y^{2}} \\ x=0 \end{array}\right. \]

9.1.8 Double integrals

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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

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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

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Problem: Calculate the surface of the area $ D $ : \[ D:\left(\frac{x}{a}+\frac{y}{b}\right)^{3}=\frac{x y}{c^{2}} \text {. } \]

9.1.12 Double integrals

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Problem: Find the double integral of the given function $ f(x, y) $ in the given area $ G $ : \[ \begin{array}{l} f(x, y)=e^{-\left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}} \\ G:\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2} \geq 1, \quad \iint_{G} f(x, y) d x d y-? \end{array} \]

9.1.13 Double integrals

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Problem: Change the order of integration in iterated integral: \[ \int_{-1}^{1} d x \int_{x^{2}}^{2-x^{2}} f(x, y) d y \]

9.1.14 Double integrals

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Problem: Find the double integral over the given area: \[ I=\iint_{D} e^{-x^{2}-y^{2}} d x d y, \quad D:\left\{\begin{array}{l} x=\sqrt{1-y^{2}} \\ x=y \\ x=-y \sqrt{3} \end{array}\right. \]

9.1.15 Double integrals

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