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3) Problem: Write down the integral sum for the function \( f(x)=3 x+1 \) on the segment \( [a, b]=[-1,2] \).

9.3.3.8 Definite Integrals

1.19 $

(3) Problem: Using the definite integral, calculate the limit of the sequence \[ S_{n}=\frac{1^{4}+2^{4}+\cdots+n^{4}}{n^{5}} \quad \text { when } n \rightarrow \infty \]

9.3.3.9 Definite Integrals

1.91 $

Problem: Prove the equality: \[ \int_{0}^{1} x^{m}(1-x)^{n} d x=\int_{0}^{1}(1-x)^{m} x^{n} d x . \]

9.3.3.10 Definite Integrals

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3) Problem: Calculate the definite integral: \[ \int_{5 / 2}^{10 / 3} \frac{\sqrt{x+2}+\sqrt{x-2}}{(\sqrt{x+2}-\sqrt{x-2})(x-2)^{2}} d x \]

9.3.3.11 Definite Integrals

2.87 $

(3) Problem: Calculate the definite integral: \[ \int_{0}^{\sqrt{3}} \frac{d x}{\sqrt{\left(4-x^{2}\right)^{3}}} \]

9.3.3.12 Definite Integrals

1.91 $

3) Problem: Find the value of the definite integral: \[ \int_{\ln 3}^{\ln 8} \frac{d x}{\sqrt{e^{x}+1}} \]

9.3.3.13 Definite Integrals

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3o Problem: Calculate the definite integral: \[ \int_{0}^{1} \frac{\sin ^{-1} x}{\sqrt{1+x}} d x . \]

9.3.3.14 Definite Integrals

0.72 $

(2) Problem: Find the value of the definite integral: \[ \int_{0}^{\sqrt{5}} \sqrt{15-3 \mathrm{x}^{2}} d x \]

9.3.3.15 Definite Integrals

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(3) Problem: Calculate the integral: \[ \int_{-\pi}^{\pi} \sqrt[3]{\sin x+5 \operatorname{tg} \frac{x}{3}} d x \]

9.3.3.16 Definite Integrals

0.72 $

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

2.87 $

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

1.19 $

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

1.67 $

食 Problem: The center of a moving circular disc moves along the major axis of an ellipse, with the plane of the disc being perpendicular to the plane of the ellipse, and the radius of the circle at each moment being equal to the ordinate of the ellipse. Calculate the volume of the solid that is obtained when the circular disc moves from one vertex to another.

9.5.7 Volume of a solid

1.43 $

震 Problem: Calculate the volume of the solid of revolution obtained by the rotation of the figure \( x^{2}+(y-2)^{2}=4 \) around the axis 0X.

9.6.5 Volume of a solid of revolution

1.19 $

30 Problem: Calculate the surface integral of the \( 1^{\text {st }} \) kind: \[ \iint_{\sigma}(3-2 z) d \sigma, \] where \( \sigma:\left\{\begin{array}{c}z=1-\frac{y^{2}}{2} \\ y=x, x=0, \quad z=0\end{array}\right. \)

9.8.10 Surface integrals

1.67 $

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