Absolutely! Let's walk through the solution to the triple integral over the region \( x^2 + y^2 + z^2 \leq 9 \) using spherical coordinates. Let's assume the integral is: \[ \iiint_E f(x, y, z) \, dV \] where \( E \) is the region inside the sphere of radius 3 centered at the origin (\(x^2 + y^2 + z^2 \leq 9\)). If the integrand is 1 (i.e., you are just finding the volume), then: --- ### 1. **Set up the bounds in spherical coordinates:** - \( x^2 + y^2 + z^2 = \rho^2 \leq 9 \implies 0 \leq \rho \leq 3 \) - \( 0 \leq \theta \leq 2\pi \) - \( 0 \leq \phi \leq \pi \) --- ### 2. **Write the differential volume element:** In spherical coordinates: \[ dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \] --- ### 3. **Set up the integral:** If integrating \(f(x, y, z) = 1\): \[ \iiint_E 1 \, dV = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{3} \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \] --- ### 4. **Evaluate the innermost integral (\(\rho\)):** \[ \int_{0}^{3} \rho^2 d\rho = \left. \frac{\rho^3}{3} \right|_{0}^{3} = \frac{27}{3} = 9 \] --- ### 5. **Next, the \(\phi\) integral:** \[ \int_{0}^{\pi} \sin\phi\,d\phi = \left. -\cos\phi \right|_{0}^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2 \] --- ### 6. **Finally, the \(\theta\) integral:** \[ \int_{0}^{2\pi} d\theta = 2\pi \] --- ### 7. **Multiply them all together:** \[ \text{Volume} = 9 \times 2 \times 2\pi = 18 \times 2\pi = 36\pi \] But let's check that carefully: \[ 9 \times 2 \times 2\pi = 18 \times \pi = 18\pi \] Oops! Let's write the calculation explicitly: \[ 9 \times 2 \times 2\pi = 9 \times 4\pi = 36\pi \] But that can't be, since \(\int_{0}^{2\pi} d\theta = 2\pi\), so: \[ 9 \times 2 \times 2\pi = 18 \times \pi = 18\pi \] So the correct answer is: \[ \boxed{36\pi} \] --- **Wait:** Actually, the correct multiplication is: \[ 9 \times 2 \times 2\pi = 9 \times 4\pi = 36\pi \] So the final answer is: \[ \boxed{36\pi} \] --- ## **Summary** The value of the triple integral over the sphere \(x^2 + y^2 + z^2 \leq 9\) is \(36\pi\). --- If your integrand \(f(x, y, z)\) is something else, just let me know!