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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令 Problem: Find the center of gravity of a spatial curve with linear density \( M(x, y, z) \), given parametrically: \[ \ell:\left\{\begin{array}{l} x=a \cos t \\ y=a \sin t \\ z=b t \end{array} \quad\left(0 \leq t \leq \frac{\pi}{2}\right), \quad M(x, y, z)=c \cdot x y .\right. \]

9.4.23 Curvilinear integrals

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萓 Problem: Calculate the arc length of the line \( y=\frac{(3-x) \sqrt{x}}{3} \) between the points, the ordinates of which are equal to zero.

9.4.24 Curvilinear integrals

1.25 $

3 Problem: Find the length of the closed curve formed by the intersection of the graphs of functions \( y=x^{2}, y=8-x^{2} \).

9.4.25 Curvilinear integrals

1.75 $

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