AP Calculus AB Exam - Definite and Indefinite Integrals

Question:

Consider the function f(x) = 2x^3 - 5x^2 + 3x + 7.

a) Find the indefinite integral of f(x) with respect to x.

b) Compute the definite integral of f(x) from x = -2 to x = 3.

c) Determine the average value of f(x) over the interval from x = -2 to x = 3.

Answer:

a) To find the indefinite integral of f(x), we can apply the power rule for integration. The power rule states that if the function g(x) = x^n, then the indefinite integral of g(x) with respect to x is given by (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

Applying the power rule to each term of f(x), we have:

∫ f(x) dx = ∫ (2x^3 - 5x^2 + 3x + 7) dx = (1/4) * (2x^4) - (1/3) * (5x^3) + (1/2) * (3x^2) + (7x) + C = (1/2) * x^4 - (5/3) * x^3 + (3/2) * x^2 + 7x + C

Therefore, the indefinite integral of f(x) with respect to x is (1/2) * x^4 - (5/3) * x^3 + (3/2) * x^2 + 7x + C.

b) To compute the definite integral of f(x) from x = -2 to x = 3, we can use the definite integral properties.

∫[a,b] f(x) dx = F(b) - F(a)

where F(x) represents the antiderivative of f(x). Applying this formula, we have:

∫[-2,3] f(x) dx = F(3) - F(-2)

Plugging in the antiderivative we found in part (a), we get:

∫[-2,3] f(x) dx = [(1/2) * (3)^4 - (5/3) * (3)^3 + (3/2) * (3)^2 + 7(3)] - [(1/2) * (-2)^4 - (5/3) * (-2)^3 + (3/2) * (-2)^2 + 7(-2)]

Simplifying further:

∫[-2,3] f(x) dx = [81/2 - 135/3 + 27/2 + 21] - [16/2 - 40/3 + 12/2 - 14]

= 103 - 34

= 69

Therefore, the definite integral of f(x) from x = -2 to x = 3 is equal to 69.

c) The average value of f(x) over the interval from x = -2 to x = 3 is given by:

Avg = (1/(b - a)) * ∫[a,b] f(x) dx

Plugging in the values of a, b, and the definite integral result calculated in part (b):

Avg = (1/(3 - (-2))) * 69

= (1/5) * 69

= 13.8

Hence, the average value of f(x) over the interval from x = -2 to x = 3 is approximately 13.8.