Cracking The Ap Calculus ab Exam 2018

examples of that here. There is one aspect of the first Fundamental Theorem, however, that involves the
area between curves (we’ll discuss that in Chapter 16).

These   theorems    are sometimes   taught  in  reverse order.

But for now, you should know the following:

If  we  have    a   point   c   in  the interval    [a, b], then

f(x)    dx  +    f(x)   dx  =    f(x)   dx

In other words, we can divide up the region into parts, add them up, and find the area of the region based
on the result. We’ll get back to this in the chapter on the area between two curves.

The second theorem tells us how to find the derivative of an integral.

Example 3: Find cost dt.

The second Fundamental Theorem says that the derivative of this integral is just cos x.

Example 4: Find (1 − t^3 ) dt.

Here, the theorem says that the derivative of this integral is just (1 − x^3 ).

Isn’t this easy? Let’s add a couple of nuances. First, the constant term in the limits of integration is a
“dummy term.” Any constant will give the same answer. For example,

(1  −   t^3 )   dt  =    (1 −   t^3 )   dt  =    (1 −   t^3 )   dt  =   1   −   x^3

In other words, all we’re concerned with is the variable term.

Second, if the upper limit is a function of x, instead of just plain x, we multiply the answer by the
derivative of that term. For example,

(1  −   t^3 )   dt  =   [1  −   (x^2 )^3 ](2x)  =   (1  −   x^6 )(2x)

Example 5: Find (t + 4t^2 ) dt = 3x^4 + 4(3x^4 )^2 .