So we obvioulsy we can reverse the process , to go from dy/dx to y , a process called integration .
Note since the differential of 2x + 1 and 2x is the same when we integrate we dont which it is so we put the answer as 2x+k where k can be 0 or any other constant . So the integral of axexpn is ax exp (n+1) / (n+1 ) .
For eg the integral of 2xexp3 is 2xexp4/4 + k = xexp4/2 + k When we integrate the line y= 2 we get 2x . This is interesting , 2 times x gives us area under the line .
If we integrate say when x= 2 to x = 4 we get the area shaded if we subtract one integration from another . Note subtracting gets rid of the constant
The same is true of curves with higher powers of x . So to get an area we integrate within limits . For example the area under the curve y = x exp 2 +1 between x = 1 to x = 2 is
First integrate we get xexp3/3 + k1+ x+k2 or since k1+k2 is a constant we get xexp3/3+x+k. so when x = 1 this has the value 1/3 + 1+k or 4/3 + k and when x = 2 it is 8/3+2+k or 14/3 +k . Subtracting 14/3+k-(4/3+k) we get 14/3 +k -4/3 -k = 10/3 units .
Integration is an incredibly useful way of getting areas under a curve when the curves are quite complicated.