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3) Problem: Calculate the surface integral of the \( 1^{\text {st }} \) kind: \[ \mathrm{I}=\iint_{S}\left(x^{2}+y^{2}\right) d s \] where \( S \) is the part of the conical surface \( z=\sqrt{x^{2}+y^{2}} \), distinguished by the condition \( z \leq 1 \).

9.8.7 Surface integrals

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Problem: Calculate the surface integral of the \( 2^{\text {nd }} \) kind: \[ \iint_{S} x^{3} d y d z+y^{3} d z d x \] where \( S \) is the outside of the ellipsoid part \[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1, \quad z \geq 0 . \]

9.8.8 Surface integrals

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3) Problem: Calculate the surface integral of the \( 2^{\text {nd }} \) kind: \[ \iint_{S}(5 x+y) d y d z+z d x d y \] where \( S \) is the outer side of the full surface of the cone \( x^{2}+y^{2} \leq z^{2}, 0 \leq z \leq 4 \).

9.8.9 Surface integrals

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Problem: Calculate the line integral of the 2nd kind. \[ \int_{\Gamma} y d x-\left(y+x^{2}\right) d y \] where \( \Gamma \) is the arc of the parabola \( y=2 x-x^{2} \) from point \( A(2 ; 0) \) to point \( B=(0 ; 0) \).

9.4.13 Curvilinear integrals

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(3) Problem: Having checked that the integrand is a total differential, calculate the line integral of the \( 2^{\text {nd }} \) kind. \[ \int_{(0,0)}^{(1,1)} x\left(1+6 y^{2}\right) d x+y\left(1+6 x^{2}\right) d y \]

9.4.14 Curvilinear integrals

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31 Problem: Calculate the line integral of the \( 1^{\text {st }} \) kind along the space curve: \[ \int_{\Gamma}\left(x^{2}+y^{2}+z^{2}\right) d l \] where \( \Gamma \) is the first turn of the helix: \[ \left\{\begin{array}{c} x=a \cos t \\ y=a \sin t, \quad a>0, b>0 . \\ z=b t \end{array}\right. \]

9.4.15 Curvilinear integrals

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Problem: Calculate the line integral of the \( 1^{\text {st }} \) kind: a) \( \int_{L}(x+y) d l \), where \( L: A B C \), b) \( \int_{L} x y d l \), where \( L:\left\{\begin{array}{c}\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \\ x \geq 0, \quad y \geq 0\end{array}\right. \).

9.4.16 Curvilinear integrals

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Problem: Calculate the first kind of line integrals along curves given parametrically: a) \( \int_{L}(x+z) d l \), where \( L:\left\{\begin{array}{c}x=t \\ y=\frac{3 t^{2}}{\sqrt{2}}, \quad 0 \leq t \leq 1, \\ z=t^{2}\end{array}\right. \) b) \( \int_{L} \frac{d l}{x^{2}+y^{2}+z^{2}} \), where \( \quad L:\left\{\begin{array}{c}x=a \cos t \\ y=a \sin t \text { is the } 1 \text { st turn. } \\ z=b t\end{array}\right. \)

9.4.17 Curvilinear integrals

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(3) Problem: Calculate the line integrals of the second kind: a) \[ \begin{array}{l} \int_{A B}\left(x^{2}-2 x y\right) d x \\ +\left(2 x y+y^{2}\right) d y \quad \text { where } A B: y \\ =x^{2}, \quad A(1 ; 1), \quad B(2,4), \end{array} \] b) \( \oint_{L}(x-y) d x+(x-y) d y, \quad L: x^{2}+y^{2}=4 \).

9.4.18 Curvilinear integrals

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31 Problem: Calculate the static moments of the curve \( L \) relative to the coordinate axes, where \( L \) is the first quarter of the circular disc of radius \( R \).

9.4.19 Curvilinear integrals

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(a) Problem: Calculate the line integral of the \( 1^{\text {st }} \) kind along a space curve: \[ \begin{array}{l} \int_{\Gamma} z d l, \quad \text { where } \Gamma \text { is the arc of a conical helix: } \\ \left\{\begin{array}{c} x=t \cos t \\ y=t \sin t, \\ z=t \end{array}\right. \end{array} \]

9.4.20 Curvilinear integrals

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(3) Problem: Calculate the line integral of the 2 nd kind: \[ \int_{\Gamma} x d y+2 y d x \] where \( \Gamma \) is the contour, formed by the lines \( y= \) \( 0, y=x, y=\sqrt{1-x^{2}} \) with a positive direction of traversal.

9.4.21 Curvilinear integrals

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Problem: Having checked that the integrand is a total differential, calculate the integral: \[ \int_{(1,1,1)}^{(2,3,4)}\left(2 x y+y^{2}+y z^{2}\right) d x+\left(x^{2}+2 x y+x z^{2}\right) d y+2 x y z d z \]

9.4.22 Curvilinear integrals

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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

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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

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