Integral Calculus help for Heat Transfer Problem

The Heat Transfer Coefficient : h = q / A * delta T

q = heat flow , cal/sec

h = heat transfer coefficient , cal/sec / M^2 Centigrade

A = heat transfer surface area M^2

delta T = difference in temp between solid surface and surrounding fluid

Consider a 100 cc sphere of ice at 0 C in 1 liter of water at 30 C insulated from

surroundings. The water contains 30 000 cal and the ice will absorb 8000 cal

melting so final temp is 22000 cal/1100 g water = 20 C

The initial surface area of ice = .01034 M^2 radius = 2.87 cm

For water in free convection ,h = 14.3 cal/sec/ M^2 * C

delta T = 30 C

Just using initial conditions , q = h * A * delta T

q = 4.5 cal/sec

8000 cal/ 4.5 cal/ sec = .5 hour for melting time

But in this problem the surface area, A is going from .01034 M^2 --> 0

and delta T is going from 30 C --> 20 C

So in the calculation would A = integral A initial - A final ?

And how would you calculate delta T ?