The area of a region/The area of a region

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T Problem: Calculate the area of the domain \( D \), bounded by the given lines: \( D:\left\{\begin{array}{c}y=(x+4) e^{-x} \\ y=0, \quad x=0\end{array}\right. \)

9.7.1 The area of a region

0.73 $

(a) Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{c} y=\sqrt{18-x^{2}} \\ y=3 \sqrt{2}-\sqrt{18-x^{2}} \end{array}\right. \]

9.7.2 The area of a region

1.7 $

3 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{c} y=\frac{2}{x}, \quad y=2 \sqrt{x} \\ y=1 \end{array}\right. \]

9.7.3 The area of a region

0.97 $

35 Problem: Calculate the area of the figure bounded by the cardioid \( r=3(1+\cos \varphi) \).

9.7.6 The area of a region

1.21 $

30 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{l} y=x^{3} \\ x+y=2 \\ y=4 \end{array}\right. \]

9.7.4 The area of a region

0.97 $

Problem: Calculate the area of the domain \( D \), bounded by the given lines: \( D:\left\{\begin{array}{l}y=x^{3} \\ y=4, \quad y=x\end{array}\right. \)

9.7.5 The area of a region

1.21 $

(1) Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{l} y^{2}-6 y+x^{2}=0 \\ y=\sqrt{3} x, \quad x=0 \end{array}\right. \]

9.7.7 The area of a region

1.46 $

30 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{c} y=\frac{1}{4} x^{2} \\ y=3 x-\frac{1}{2} x^{2} \end{array}\right. \]

9.7.8 The area of a region

0.97 $

30 Problem: Calculate the area of triangle \( A B C \) with the coordinates of vertices: \( A(2 ; 3), B(7 ; 2), C(4 ; 5) \).

9.7.9 The area of a region

1.21 $

3 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{l} y^{2}+2 y-3 x+1=0 \\ 3 x-3 y-7=0 \end{array}\right. \]

9.7.10 The area of a region

0.97 $

T Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D: 2 x+3 y^{2}=0, \quad 2 x+2 y+1=0 . \]

9.7.11 The area of a region

1.21 $

3 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D: y=20-x^{2}, y=-8 x . \]

9.7.12 The area of a region

0.97 $

3) Problem: Calculate the area of a figure bounded by the lemniscate \( \rho^{2}=2 \cos 2 \varphi \) and the circle \( \rho=1 \), and located outside the circle.

9.7.13 The area of a region

2.43 $

(3) Problem: Find the area of the surface formed by the rotation of the cardioid \[ x=a(2 \cos t-\cos 2 t), \quad y=a(2 \sin t-\sin 2 t) \] around the axis \( O X(0 \leq t \leq \pi) \).

9.7.14 The area of a region

1.94 $

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

9.7.15 The area of a region

0 $

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