So long as the curve is traversed (I always hated that word; travelled sounds better) once, then you can use direct substitution as you have done. What is the answer the book gets?
The diagram shows the curve defined by parametric equations ,
for values of t between -3 and 3 find the shaded area.
I have drawn the diagram shows in my book and this my set up,
However when I compute this, I do not get the right answer. Is this the correct method?