2000 Calculus AB Free-Response: Question 1 Part B

Lets continue now with Question 1 Part B.

Part be asks that we revolve f(x) and g(x) around the x-axis. To do this, must integrate f(x)^2 - g(x)^2 using the same parameters as Part A (0 to .94194). Then, after it is integrated, we must multiply it by pi.

Remember that the first part of the AP Calculus free response allows you to use a graphing calculator. A graphing calculator is actually required to solve this problem because the indefinite integral of e^(-x^2) can not be found.

Back to the problem... just put in the definite integral in your calculator and press enter. If you have a TI-83 Plus or TI-84 Plus, type fnInt((e^((-2x)^2)-((1-cos(x))^2),x,0,.94194).

You should get Volume = 1.7466 units^3

2000 Calculus AB Free-Response: Question 1 Part A

In my last post, Learn from the past AP Tests: 2000, I linked to the 2000 AP Calculus AB Free-Response questions. Today, I will start to go through the solutions, starting with question number one, part A.

Question number one, part A asks for the area of a region, R, between two functions (also known as curves), which I will call f(x) and g(x). f(x)=e^(-(x^2)) and g(x)=1 - cos(x)

To solve, we must first find the area under the top curve, f(x). To do this, we must integrate f(x) (Also known as taking the antiderivative. The antiderivative of f(x) can be written as F(x).). So F(x)= -(1/2)e^(-(x^2)). The easiest way to do that is u substitution.

Now, lets integrate g(x). G(x)= x - sin(x).

Now that we know the two antiderivatives, we need to find out wher f(x) intersect with g(x); so where f(x) = g(x). Remember, this is a calculator portion, so you can simply plug this into a calculator (In fact, you can solve this question in a much easier manner using your calculator. I will show you how in my next post.). The results show the intersection, B, is at .94194.

Lets plug it all in! The integral of f(x) - g(x) from A=0 to B=.94194, can also be written as [F(x) - G(x)]₀ ^.94194, which can be written as F(.94194)-F(0) - (G(.94194) - G(0)).

It all equals .591.

If you have any questions, please leave a comment.