Double integrals

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

9.1.16 Double integrals

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Problem: Change the order of integration in iterated integral: \[ I=\int_{-2}^{2} d x \int_{-\sqrt{\frac{4-x^{2}}{2}}}^{\sqrt{\frac{4-x^{2}}{2}}} f(x, y) d y \]

9.1.17 Double integrals

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Problem: Find the double integral over the given area \( S \) : \[ I=\iint_{(S)} \sqrt{a^{2}-x^{2}-y^{2}}, \] \( S:\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right), \quad x \geq 0 \).

9.1.18 Double integrals

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

9.1.19 Double integrals

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Problem: Find the double integral over the given area \( D \) : \[ \iint_{(D)}(12-2 x-3 y) d x d y, \quad D:\left\{\begin{array}{l} 12-2 x-3 y=0 \\ x=0, \quad y=0 \end{array}\right. \]

9.1.20 Double integrals

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

9.1.21 Double integrals

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Problem: Use the double integral to calculate the area of the given region \( D \) : \[ D:\left\{\begin{array}{l} x=0, y=0 \\ x+y=1 \end{array}\right. \]

9.1.22 Double integrals

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Problem: Find the double integral over the area \( D \), bounded by the indicated lines: \[ \begin{array}{l} \iint_{D}\left(24 x y+18 x^{2} y^{2}\right) d x d y, \\ D: x=1, \quad y=x^{3}, \quad y=-\sqrt[3]{x} . \end{array} \]

9.1.23 Double integrals

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Problem: Find the double integral \( \iint_{D}(1+x+y) d x d y \), where \( D- \) is the area, bounded by the parabola \( x=\sqrt{y} \) and straight lines \( y=-x, y=2 \).

9.1.24 Double integrals

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Problem: Find the double integral over the given region: \[ \iint_{D}\left(x^{3}+y^{3}\right) d x d y, \quad D:\left\{\begin{array}{ll} x=0, & x=3 \\ y=1, & y=4 \end{array}\right. \]

9.1.25 Double integrals

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Problem: Find the double integral over the given region: \[ \iint_{D}(x-y) d x d y, \quad D:\left\{\begin{array}{l} x=0, y=0 \\ x+y=1 \end{array}\right. \]

9.1.26 Double integrals

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Problem: Change the order of integration: \[ \int_{0}^{\frac{\pi}{4}} d y \int_{0}^{\sin y} f d x+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} d y \int_{0}^{\cos y} f d x \]

9.1.27 Double integrals

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Problem: Find the double integral over the given region: \[ \begin{array}{l} \iint_{D}\left(12 x y+27 x^{2} y^{2}\right) d x d y \\ D: x=1, y=x^{2}, y=-\sqrt[3]{x} \end{array} \]

9.1.28 Double integrals

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Problem: Find the double integral over the given region: \[ \begin{array}{l} \iint_{D} y e^{\frac{x y}{4}} d x d y, \\ D: y=\ln 2, y=\ln 3, x=4, x=8 . \end{array} \]

9.1.29 Double integrals

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Problem: Find the double integral over the region given in polar coordinates: \[ \iint_{(D)} r^{3} d r d a, \quad r^{2}=b^{2} \cos (2 a), a \in\left[0 ; \frac{\pi}{4}\right] . \]

9.1.30 Double integrals

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