To evaluate the triple integral

$I = \int_0^{\pi} \int_0^{2\pi} \int_1^4 \rho^4 \sin^3(\varphi) \, d\rho \, d\theta \, d\varphi,$

we can proceed by integrating each variable one at a time.

Step-by-Step Solution:

1. Integrate with respect to $\rho$:

   The inner integral is:

   $\int_1^4 \rho^4 \, d\rho.$

   Integrating $\rho^4$ from 1 to 4:

   $\int_1^4 \rho^4 \, d\rho = \left[ \frac{\rho^5}{5} \right]_1^4 = \frac{4^5}{5} - \frac{1^5}{5} = \frac{1024}{5} - \frac{1}{5} = \frac{1023}{5}.$

2. Integrate with respect to $\theta$:

   The next integral is:

   $\int_0^{2\pi} \frac{1023}{5} \, d\theta.$

   Since $\frac{1023}{5}$ is constant with respect to $\theta$, this simplifies to:

   $\frac{1023}{5} \int_0^{2\pi} d\theta = \frac{1023}{5} \cdot 2\pi = \frac{2046\pi}{5}.$

3. Integrate with respect to $\varphi$:

   The final integral is:

   $\int_0^{\pi} \frac{2046\pi}{5} \sin^3(\varphi) \, d\varphi.$

   To solve this, we use the integral formula for $\sin^3(\varphi)$:

   $\int \sin^3(x) \, dx = \int (1 - \cos^2(x)) \sin(x) \, dx.$

   However, directly integrating over $\sin^3(\varphi)$ from 0 to $\pi$ is simpler using trigonometric identities and standard integral results. The result of this integral is $\frac{4}{3}$.

   Therefore,

   $\int_0^{\pi} \sin^3(\varphi) \, d\varphi = \frac{4}{3}.$

4. Combine the results:

   Now, putting it all together:

   $I = \frac{2046\pi}{5} \cdot \frac{4}{3} = \frac{8184\pi}{15}.$

Final Answer:

$I = \frac{8184\pi}{15}.$

Would you like further details on any step?

Related Questions:

1. How does integrating with respect to different coordinates affect the outcome?
2. What is the significance of the trigonometric identity in simplifying the integral?
3. How does the range of $\rho$, $\theta$, and $\varphi$ affect the final result?
4. What are other applications of this type of spherical coordinate integration?
5. How can we interpret this integral geometrically?

Tip: Breaking down multiple integrals into single integrals by isolating each variable helps simplify complex computations.