Stuck on a Statistics 2 question!

On AQA Jan 13 question 6, I don't understand why you don't integrate 1/216(t^3), I thought the area under the curve was probability 1 and therefore to find 0.9, you integrated 1/216(t^3) and put it equals to 0.9. The mark scheme just says to put it equal to 0.9 without integrating... I am very confused.

Thanks.

Links to paper and mark scheme:

http://filestore.aqa.org.uk/subjects/AQA-MS2B-QP-JAN13.PDF
http://filestore.aqa.org.uk/subjects/AQA-MS2B-W-MS-JAN13.PDF

The capital F means it is the cumulative total so it has already been integrated meaning that you don't have to integrate as well if it was f(T) then you would have to integrate. You probably assumed it was a p.d.f

You've confused CDF and PMF. CDF is the integral of PMF. $F(t) = \displaystyle\int_{0}^t f(t)dt$.

You're given F(the CDF) in the question, so it's already been integrated.