The "Region of Integration" window shows the integral region defined by functions in spherical coordinates.  The wireframe colored according to the value of the function is displayed with the "Wire" checkbox is selected.  The surface about the wireframe colored according to the value of the function is displayed with the "Surface" checkbox is selected.  A gray surface at theta0 as well as a blue line on this surface at phi0 are both displayed with the "Slice" checkbox is selected.

The "Inner Integral" window displays in green the integrand of the inner double intergal used in calculating the mass of the object as a function of theta and phi.  When the "Guess" checkbox is selected the "GuessedInnerIntegral(theta, phi)" is displayed as a red surface.  Compare your answer with the green surface.

The "Double Integral" Window displays in orange the double integral as a function of theta that is the final integrand in evaluating the mass of the region.  When the "Guess" checkbox is selected the "GuessedDoubleIntegral(theta)" is displayed as a function of theta.  Compare your answer with the orange curve.

The readouts display the suggested integrals evaluated at theta0 and phi0 where these variables would be part of the result of integration.