Originally Posted by

**HallsofIvy** Yes, although -16.5 would be a better approximation.

One of the first things you should have learned about "definite integrals" (actually, the **definition**) is that if F is an anti-derivative of f (that is, if dF/dx= f) then $\displaystyle \int_a^b f(x)dx= F(b)- F(a)$.

Actually, yes, finding anti-derivatives is, typically harder than differentiating- although **evaluating** definite integrals should not be more difficult than evaluating any functions. There is a general distinction in mathematics between "direct problems", where we are given a specific definition or formula, and "inverse problems" where we are to "reverse" some direct problem. For example, if I told you that $\displaystyle y= x^5- 3x^4+ 4x^2- 7x+ 9$ and ask you to find y when x= 3, that would be easy- just do the arithmetic in that formula. But if I told you that y= 3 and asked you to find x,, that would be a much more difficult problem- and there might be many solutions or none.

We learn, in introductory Calculus, a formula for the derivative of a function so finding a derivative can always go back to using that formula (though we can also use the many "theorems" for special cases derived from that formula). That's a **direct** problem. But an anti-derivative of f, say, is only defined as "a function whose derivative is f". We don't have a direct formula to use, we have to try to "remember" a function whose derivative is the given function. That's the "inverse" problem.