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The question asks for the integral used to calculate the area of a rose curve, specifically for the polar equation \( \rho = 4 \cos(3\varphi) \).

### Step-by-Step Solution:

The general formula for the area enclosed by a polar curve \( \rho = f(\varphi) \) is:

\[
A = \frac{1}{2} \int_{\alpha}^{\beta} \rho^2 \, d\varphi
\]

In this case, \( \rho = 4 \cos(3\varphi) \), so:

\[
\rho^2 = (4 \cos(3\varphi))^2 = 16 \cos^2(3\varphi)
\]

Thus, the area becomes:

\[
A = \frac{1}{2} \int_{\alpha}^{\beta} 16 \cos^2(3\varphi) \, d\varphi = 8 \int_{\alpha}^{\beta} \cos^2(3\varphi) \, d\varphi
\]

For a rose curve of the form \( \rho = 4 \cos(3\varphi) \), the total area is calculated over one petal, and because the curve has symmetry, the limits of integration for one petal range from \( \varphi = -\frac{\pi}{6} \) to \( \frac{\pi}{6} \). The entire area is given by multiplying the area of one petal by the number of petals, which for this curve (with \( 3 \varphi \)) is 6 petals.

The correct integral that gives the total area is:

\[
8 \int_0^{2\pi} \cos^2(3\varphi) \, d\varphi
\]

Therefore, the correct option is:

- \( 8 \int_0^{2\pi} \cos^2(3\varphi) \, d\varphi \)

This matches the fourth option.

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Let me know if you need further details or explanations. Here are 5 additional questions to explore this concept:

1. How do you derive the formula for the area enclosed by a polar curve?
2. What is the significance of the limits of integration for different polar curves?
3. How does the number of petals affect the total area of a rose curve?
4. Can this method be applied to curves with sine functions, such as \( \rho = a \sin(k\varphi) \)?
5. How do you simplify integrals involving trigonometric functions like \( \cos^2(\theta) \)?

**Tip:** For integrals involving \( \cos^2(\theta) \), remember to use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) to simplify calculations.

Укажите интеграл, с помощью которого вычисляется площадь розы ρ = 4cos(3φ)

Learn how to calculate the area of a rose curve defined by the polar equation ρ = 4cos(3φ). This detailed step-by-step solution covers the key formulas and concepts in polar coordinates and trigonometry. Ideal for students studying university-level calculus.

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