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The problem asks for the integral that calculates the area of the rose curve given by the polar equation \( \rho = 4 \cos 3\varphi \).

To find the area \( A \) of one petal of a rose curve given by \( \rho = a \cos(n\varphi) \) or \( \rho = a \sin(n\varphi) \), the formula is:

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

For the given equation \( \rho = 4 \cos 3\varphi \):

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

Thus, the area integral becomes:

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

Since this is a three-petaled rose (because \( n = 3 \) is odd), the limits of integration for one petal range from \( \varphi = -\frac{\pi}{6} \) to \( \varphi = \frac{\pi}{6} \), or equivalently, you could integrate from \( 0 \) to \( \frac{\pi}{3} \) for one full petal.

The total area of all three petals would then be:

\[
A_{\text{total}} = 3 \times 8 \int_{0}^{\frac{\pi}{3}} \cos^2 3\varphi \, d\varphi = 24 \int_{0}^{\frac{\pi}{3}} \cos^2 3\varphi \, d\varphi
\]

So the correct answer among the options would be:

\[
\boxed{24 \int_{0}^{\frac{\pi}{3}} \cos^2 3\varphi \, d\varphi}
\]

This corresponds to the third option in the provided choices.

Would you like more details or have any further questions?

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Here are some related questions you might find helpful:

1. How do you determine the number of petals in a rose curve given by \( \rho = a \cos(n\varphi) \)?
2. What is the general formula for the area of a polar curve?
3. How does the range of integration change for different numbers of petals in a rose curve?
4. What is the difference between the integrals for \( \cos \) and \( \sin \) functions in polar coordinates?
5. How can trigonometric identities simplify the integration of \( \cos^2 \) functions?

**Tip:** When integrating trigonometric functions squared, use the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) to simplify the process.

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

Find the correct integral to calculate the area of the rose curve given by the polar equation ρ = 4cos(3φ). This solution uses key concepts in polar coordinates and trigonometric integrals, suitable for university-level calculus.

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