curves cross the x-axis because multiple integrals will be required if this happens. In this case,
the curves intersect on the x-axis at x = 2. Therefore, the area between the curves will be:
(x^3 − 2 x^2 − 5x + 6) − (x^2 − x − 6) dx + (x^2 − x − 6) − (x^3 − 2x^2 − 5x + 6) dx = 32 +
.
- C This problem will require you to be familiar with the Trapezoid Rule. This is very easy to do
on the calculator, and some of you may even have written programs to evaluate this. Even if
you haven’t, the formula is easy. The area under a curve from x = a to x = b, divided into n
intervals, is approximated by the Trapezoid Rule, and it is
[y 0 + 2y 1 + 2y 2 + 2y 3 ...+ 2yn − 2 + 2yn − 1 + yn]
This formula may look scary, but it actually is quite simple, and the AP Exam never uses a very
large value for n anyway.
Step 1: . Plugging into the formula, we get
[(1^2 + 1) + 2(1.5^2 + 1) + 2(2^2 + 1) + 2(2.5^2 + 1) + (3^2 + 1)]
This is easy to plug into your calculator and you will get 10.75 or .
Step 2: In order to find the error, we now need to know the actual value of the integral.
(x^2 + 1) dx = + x =
Step 3: The error is − = .
- C Plug in the given point, (0, 3), into the equation for the curve, y = ax^2 + bx + c, thus c = 3. Next,
rewrite the equation for the normal line in slope-intercept form, y = x + 3. Since this line is
normal to the curve, the slope of the tangent line is the opposite reciprocal to the slope of the
normal line. The slope of the tangent line is −5 to evaluate it at (0, 3). Take the derivative of y
