AP Calculus AB Exam Question

Consider the function f(x) = 4 - 3x² on the interval [-2, 3].

a) Find the exact value of the area bounded by the curve f(x) and the x-axis on the interval [-2, 3].

b) Write an integral expression, in terms of x, for the area bounded by the curve f(x), the x-axis, and the lines x = -2 and x = 3.

c) Using the integral expression in part (b), evaluate the definite integral to find the area bounded by the curve f(x), the x-axis, and the lines x = -2 and x = 3.

Answer

a) To find the exact value of the area bounded by the curve f(x) and the x-axis on the interval [-2, 3], we need to evaluate the definite integral of the function f(x) over the interval [-2, 3]. The area can be found using the formula:

where a and b are the limits of integration.

Using the given function f(x) = 4 - 3x², the integral expression for the area becomes:

To evaluate this integral, we need to split it into two separate integrals over the intervals [-2, 0] and [0, 3] due to the absolute value.

For the interval [-2, 0], the integral expression becomes:

Integrating the function f(x) = 4 - 3x² within this interval:

For the interval [0, 3], the integral expression becomes:

Integrating the function f(x) = 4 - 3x² within this interval:

To find the total area, we need to add the two calculated areas together:

Therefore, the exact value of the area bounded by the curve f(x) = 4 - 3x² and the x-axis on the interval [-2, 3] is -15.5.

b) The integral expression for the area bounded by the curve f(x), the x-axis, and the lines x = -2 and x = 3 can be written as:

c) Evaluating the definite integral to find the area bounded by the curve f(x) = 4 - 3x², the x-axis, and the lines x = -2 and x = 3:

Therefore, the area bounded by the curve f(x) = 4 - 3x², the x-axis, and the lines x = -2 and x = 3 is -15.5.