Solve spring problem with integrals

2J of work is needed to stretch a spring from its natural length of 30cm to a length of 42cm

a) how much work is needed to stretch the spring from 35cm to 40 cm?

2=INT kx dx from 0 to 12
solve for k
k=1/36

W=INT 1/36x dx from 5 to 10
= 1.04J (which is the correct answer)

b) how far beyond its natural length will a force of 30N keep the spring stretched?

since i know that k=1/36 from part a), I just did 30=(1/36)x since F= kx
x = 1080 m
but the actual answer is 10.8 cm...
what went wrong?

Physics is easier when you keep track of the units.
In the first part of the problem you correctly deduce k= 1/36, but what are the units?
Well, you have divided J by cm^2, so if you stick to the 1/36, it is 1/36 J/cm^2.

But 1 J = 1 N m = 1 N * 100 cm = 100 N cm. Hence actually

k = 1/36 * 100 N cm / cm^2 = 100/36 N/cm

Now for the second part, you have
X = F / k = 30 N / (100/36. N/cm) = 10.8 cm.

Lesson: always write the units in your calculations, it makes it easy to check that you are calculating the correct quantity.