AP Calculus AB Exam Question:

Consider the function f defined by the equation f(x) = 3x^2 + 2x + 1.

a) Find the area between the curve y = f(x) and the x-axis on the interval [-1, 2].

Solution:

To find the area under the curve, we need to integrate the function f(x) over the given interval [-1, 2]. Since the function is a quadratic, we can easily find the antiderivative and evaluate the definite integral.

First, let's find the antiderivative F(x) of f(x):

F(x) = ∫ (3x^2 + 2x + 1) dx

To find the antiderivative of each term, we can apply the power rule of integration:

F(x) = x^3 + x^2 + x + C

Next, we can evaluate the definite integral over the interval [-1, 2]:

Area = ∫[-1, 2] f(x) dx

Area = F(2) - F(-1)

Now let's substitute the limits of integration into F(x):

Area = (2^3 + 2^2 + 2) - ((-1)^3 + (-1)^2 + (-1))

Simplifying further:

Area = (8 + 4 + 2) - (-1 + 1 - 1)

Area = 14

Therefore, the area between the curve y = f(x) and the x-axis on the interval [-1, 2] is equal to 14 square units.