Water flows into a tank according to the rate F(t)= (t+6)/(1+t), and at the same time empties out at the rate E(t)= (ln(t+2))/(t
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The curve 
is parameterized by
so in the line integral, we have
You are mistaken in thinking that the gradient theorem applies here. Recall that for a scalar function
, we have gradient 
. The theorem itself then says that the line integral of 
along a curve 
parameterized by 
, where 
, is given by
Specifically, in order for this theorem to even be considered in the first place, we would need to be integrating with respect to a vector field.
But this isn't the case: we're integrating
, a scalar function.
so in the line integral, we have
You are mistaken in thinking that the gradient theorem applies here. Recall that for a scalar function
Specifically, in order for this theorem to even be considered in the first place, we would need to be integrating with respect to a vector field.
But this isn't the case: we're integrating