Question

Use differentials to estimate the amount of paint needed to apply a 1/4 cms thick coat of paint to a sphere having diameter 4 cms.

Answer 1

Geometry?
Find the volume of a sphere with diamete 4.5cm and subtract the volume of a sphere with diameter 4cm, and that will be the amount of paint, in cm cubed, you will need.

Answer 2

no, this is surface area

Answer 3

If you need to solve this using differentials (like you mentioned to me in chat), you will want to take a look at the surface area of the sphere. As if you painted the sphere over and over again until you had a 1/4 cm thick coating, you will want to find the total amount of surface areas added up in this process.
The equation for surface area of a sphere is:
\[A = 4\pi r^2\]
So from this, you will want to first convert the diameters to radii; 4cm to 2cm, and 4.5cm (the diameter plus 0.25cm paint coating on each side) to 2.25cm. Then, you should solve this equation:
\[\int\limits_{2}^{2.25} 4\pi r^2\]
\[4\pi\int\limits\limits\limits_{2}^{2.25}r^2 = 4\pi \left[ (1/3)r^3 \right]_{2}^{2.25} = 4\pi \left[ (1/3)r^3 \right]_{2}^{2.25} = (4/3)\pi \left[ r^3 \right]_{2}^{2.25} = (4/3)\pi (2.25^3 - 2^3) \]\[\approx 14.2026\]
You'll notice that solving that incredibly time-consuming integral essentially did the exact same thing as I mentioned before; by subtracting the without-paint-volume from the with-paint-volume.