To convert the given triple integral from rectangular coordinates to spherical coordinates, let's break the process into steps.

Original Integral:

$\int_{-7}^{7} \int_{0}^{\sqrt{49 - y^2}} \int_{-\sqrt{49 - x^2 - y^2}}^{\sqrt{49 - x^2 - y^2}} -4(x^2 + y^2 + z^2)^2 \, dz \, dx \, dy$

1. Recognize the bounds in rectangular coordinates:

• $x^2 + y^2 + z^2 = 49$: This represents a sphere with radius $7$.
• The limits on $x, y, z$ describe a solid sphere centered at the origin with radius $7$.

2. Express the integrand in spherical coordinates:

In spherical coordinates:

• $x = \rho \sin\phi \cos\theta$,
• $y = \rho \sin\phi \sin\theta$,
• $z = \rho \cos\phi$,
• $x^2 + y^2 + z^2 = \rho^2$,
• The volume element is $dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$.

The integrand $-4(x^2 + y^2 + z^2)^2$ becomes: $-4(x^2 + y^2 + z^2)^2 = -4\rho^4$

3. Determine the bounds in spherical coordinates:

• $\rho$: Radius ranges from $0$ to $7$ (since the sphere's radius is 7).
• $\phi$: Polar angle ranges from $0$ to $\pi$ (full vertical sweep).
• $\theta$: Azimuthal angle ranges from $0$ to $2\pi$ (full horizontal sweep).

4. Write the integral in spherical coordinates:

Substitute the integrand and bounds into the spherical coordinate integral:

$\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{7} -4\rho^4 \cdot \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$

Simplify the volume element $\rho^4 \cdot \rho^2$: $\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{7} -4\rho^6 \sin\phi \, d\rho \, d\phi \, d\theta$

Final Expression:

The integral in spherical coordinates is: $\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{7} -4\rho^6 \sin\phi \, d\rho \, d\phi \, d\theta$

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Would you like me to clarify any step? Here are five questions to consider:

1. Would you like a step-by-step explanation of the coordinate transformation rules?
2. Should I elaborate on the calculation of volume elements?
3. Do you need further clarification on the bounds in spherical coordinates?
4. Would you like examples of converting other integrals to spherical coordinates?
5. Do you need help visualizing this region geometrically?

Tip: When converting integrals to spherical coordinates, ensure that the symmetry of the region is fully utilized to simplify bounds and computations.