Interpretations and properties of definite integrals - AP Calculus AB

Here we are using several basic properties of definite integrals as well as the fundamental theorem of calculus.

First, you can pull coefficients out to the front of integrals.

Second, we notice that our lower bound is bigger than our upper bound. You can switch the upper and lower bounds if you also switch the sign.

Lastly, our integral "distributes" over addition and subtraction. That means you can split the integral by each term and integrate each term separately.

Now we integrate and calculate using the Fundamental Theorem of Calculus.

We proceed as follows

. (Start)

. (Break up the integral using the additive rule.)

. (We don't have information about the 2nd integral, so we solve our first equation for and replace it in this integral.)

. (Factor out the using linearity.)

. (Substitute in what we were given.)

.

You are provided f(1) and are told to find the value of f(4). By the FTC, the following follows:

(integral from 1 to 4 of f'(x)dx) + f(1) = f(4)

16 + 12 = 28

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n2 + 6)

Use L'Hopitals rule to find the limit.

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n2 + 6)

lim as n approaches infiniti of ((12n2) – 6)/((3n2) – 4n + 6)

lim as n approaches infiniti of 24n/(6n – 4)

lim as n approaches infiniti of 24/6

The limit approaches 4.