Curvilinear integrals

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(3) Problem: Find the arc length of the curve which is specified parametrically: \[ L: r=2 \cos ^{3} \frac{\varphi}{3}, \varphi \in\left[0 ; \frac{3 \pi}{2}\right] \]

9.4.1 Curvilinear integrals

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(3) Problem: Find the mass of the curve, given parametrically: \( \ell: y=\ln \cos x, x \in\left[0 ; \frac{\pi}{3}\right] \), density \( \rho=e^{y} \).

9.4.2 Curvilinear integrals

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(a) Problem: Calculate the line integral from point \( A \) to point \( C \) along three different paths: \[ \begin{array}{l} A(0 ; 1), \quad B(3 ; 1), \quad C(3 ; 10) \\ I=\int_{\ell}(2 y+x) d x+(2 x-1) d y \end{array} \]

9.4.3 Curvilinear integrals

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T Problem: Calculate the line integral of the 2nd kind using Green's formula. \[ \oint_{C}\left(x^{3}-2 y\right) d x+\left(y^{3}-x\right) d y \] where \( C: x=1, y^{2}=x \).

9.4.11 Curvilinear integrals

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(3) Problem: Calculate the line integral: \[ \int_{L}\left(x^{5}+8 x y\right) d L, \text { where } L: 4 y=x^{4}, 0 \leq x \leq 1 . \]

9.4.4 Curvilinear integrals

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3 Problem: Calculate the line integral: \[ \int_{L} \sqrt{\frac{a^{2}}{b^{2}} y^{2}+\frac{b^{2}}{a^{2}} x^{2}} d L \] where \( L:\left\{\begin{array}{l}x=a \cos t \\ y=b \sin t\end{array}, \quad 0 \leq t \leq \frac{\pi}{2}\right. \).

9.4.5 Curvilinear integrals

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(2) Problem: Calculate the line integral: \[ \int_{L}(x+y) d L, \text { where } L \text { is the petal of the } \] lemniscate located in the first quadrant.

9.4.6 Curvilinear integrals

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3) Problem: Calculate the line integral using the polygonal \( L \) : \[ \int_{L}\left(x^{3}+y\right) d x+\left(x+y^{2}\right) d y \] where 1) \( L: A B C, 2) L: A B C A, A(7 ; 7), B(3 ; 7), C(3 ; 5) \).

9.4.7 Curvilinear integrals

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Problem: Calculate the line integral: \( L:\left\{\begin{array}{c}x=t^{2} \\ y=t\end{array}, \quad 1 \leq t \leq 2, \quad \int_{L} y x d x+y^{2} d y\right. \).

9.4.8 Curvilinear integrals

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(3) Problem: Calculate the line integral over the closed polygonal \( A B C D A \) using Ostrogradsky-Green formula: \[ \begin{array}{l} A(3 ; 2), \quad B(6 ; 2), \quad C(6 ; 4), \quad D(3 ; 4) \\ \oint_{L} \sqrt{x^{2}+y^{2}} d x+y\left[x y+\ln \left(x+\sqrt{x^{2}+y^{2}}\right)\right] d y \end{array} \]

9.4.9 Curvilinear integrals

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T Problem: Calculate the arc length of the curve given in Cartesian coordinates: \[ x=\ln \cos y, \quad 0 \leq y \leq \frac{\pi}{3} . \]

9.4.10 Curvilinear integrals

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3 Problem: Calculate the arc length of the curve given in polar coordinates: \[ r=3(1+\sin a),-\frac{\pi}{6} \leq a \leq 0 \]

9.4.12 Curvilinear 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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