You are correct that the angle \( \theta \) should range from \( 0 \) to \( 2\pi \), as the curve \( r = 5\cos\theta \) traces a complete cardioid over this interval. Let's correct the computation with this range.

### Corrected Approach:

### Step 1: Set up the integral
The volume integral in cylindrical coordinates remains the same:
\[
V = \int_{\theta=0}^{2\pi} \int_{r=0}^{5\cos\theta} \int_{z=0}^{-2r\sin\theta} r \, dz \, dr \, d\theta
\]

### Step 2: Integrate with respect to \( z \):
\[
\int_{z=0}^{-2r\sin\theta} r \, dz = r[-2r\sin\theta] = -2r^2\sin\theta
\]

### Step 3: Integrate with respect to \( r \):
\[
\int_{r=0}^{5\cos\theta} -2r^2\sin\theta \, dr = -2\sin\theta \int_{0}^{5\cos\theta} r^2 \, dr = -2\sin\theta \cdot \frac{(5\cos\theta)^3}{3} = -\frac{250\cos^3\theta\sin\theta}{3}
\]

### Step 4: Integrate with respect to \( \theta \):
Now, we integrate from \( \theta = 0 \) to \( 2\pi \):
\[
V = \int_{0}^{2\pi} -\frac{250\cos^3\theta\sin\theta}{3} \, d\theta
\]

### Step 5: Solve the integral
Notice the integrand \( -250\cos^3\theta\sin\theta \) is an odd function of \( \sin\theta \) (since it’s multiplied by \( \cos^3\theta \)). When integrated over a full period \( [0, 2\pi] \), the positive and negative contributions cancel out.

Thus, the volume is:
\[
V = 0
\]

### Conclusion:
The corrected volume of the solid is indeed \( \boxed{0} \), which occurs due to the symmetry of the function over the interval \( [0, 2\pi] \). The positive and negative contributions to the volume cancel each other out.

Do you have any more questions, or would you like to dive into a different aspect of the problem?

### Related Questions:
1. What role does symmetry play in definite integrals?
2. How do you recognize odd and even functions in integrals?
3. What is the significance of the bounds in polar coordinates?
4. How does changing bounds affect the outcome of integrals?
5. What are common cases where volume integrals yield zero?

### Tip:
Always consider the symmetry and periodicity of the integrand when computing volumes or areas, especially in problems involving trigonometric functions.

Explore how to calculate the volume of a solid using cylindrical coordinates and volume integrals. Understand the role of symmetry and periodicity in canceling out contributions to the volume over the interval [0, 2π]. Detailed step-by-step solution and mathematical insights provided.

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