AP Calculus AB: Fundamental Theorem of Calculus Part 2 and Properties (Drill 1)

Question 1. What is the value of \( \int_0^3 (x^2 + 2)\,dx \)?

• A) \( 9 \)
• B) \( 11 \)
• C) \( 15 \) ✓
• D) \( 6 \)

Explanation: Choice C is correct. The antiderivative of \( x^2 + 2 \) is \( F(x) = \dfrac{x^3}{3} + 2x \). By FTC Part 2: \( F(3) - F(0) = \left(\dfrac{27}{3} + 6\right) - 0 = 9 + 6 = 15 \). Choice A is incorrect because the student computed only \( \int_0^3 x^2\,dx = 9 \), omitting the integral of the constant term \( 2 \). Choice B is incorrect because the student evaluated the antiderivative at the upper limit as \( \dfrac{27}{3} + 2 = 11 \), treating \( 2x \) as the constant \( 2 \) rather than \( 2(3) = 6 \). Choice D is incorrect because the student computed only \( \int_0^3 2\,dx = 6 \), ignoring the \( x^2 \) term entirely.

Question 2. If \( \int_2^7 f(x)\,dx = 10 \), what is the value of \( \int_7^2 f(x)\,dx \)?

• A) \( -10 \) ✓
• B) \( 0 \)
• C) \( 10 \)
• D) Cannot be determined without knowing \( f(x) \)

Explanation: Choice A is correct. Reversing the limits of integration negates the integral: \( \int_7^2 f(x)\,dx = -\int_2^7 f(x)\,dx = -10 \). Choice B is incorrect because \( \int_7^2 f(x)\,dx = 0 \) would require \( \int_2^7 f(x)\,dx = 0 \), which is not given. Choice C is incorrect because it ignores the sign change that occurs when limits are reversed, the most common error on this type of question. Choice D is incorrect because the reversal property holds for any integrable function; no additional information about \( f \) is needed.

Question 3. Suppose \( \int_1^5 f(x)\,dx = 14 \) and \( \int_1^3 f(x)\,dx = 6 \). What is the value of \( \int_3^5 f(x)\,dx \)?

• A) \( 20 \)
• B) \( -8 \)
• C) \( \dfrac{7}{3} \)
• D) \( 8 \) ✓

Explanation: Choice D is correct. By the interval additivity property: \( \int_1^5 f(x)\,dx = \int_1^3 f(x)\,dx + \int_3^5 f(x)\,dx \), so \( \int_3^5 f(x)\,dx = 14 - 6 = 8 \). Choice A is incorrect because the student added the two given values rather than subtracting: \( 14 + 6 = 20 \). Choice B is incorrect because the student reversed the subtraction, computing \( 6 - 14 = -8 \). Choice C is incorrect because the student divided the two given integrals \( \left(\dfrac{14}{6} = \dfrac{7}{3}\right) \), which has no valid interpretation under any integral property.

Question 4. If \( \int_0^4 f(x)\,dx = 10 \) and \( \int_0^4 g(x)\,dx = -3 \), what is the value of \( \int_0^4 [3f(x) - 2g(x)]\,dx \)?

• A) \( 24 \)
• B) \( 36 \) ✓
• C) \( 21 \)
• D) \( 27 \)

Explanation: Choice B is correct. By linearity: \( 3\int_0^4 f(x)\,dx - 2\int_0^4 g(x)\,dx = 3(10) - 2(-3) = 30 + 6 = 36 \). Choice A is incorrect because the student treated \( g(x) \) as positive, computing \( 3(10) - 2(3) = 30 - 6 = 24 \), a sign error when applying the coefficient to a negative integral value. Choice C is incorrect because the student applied the single coefficient 3 to the sum of both integral values rather than distributing the separate coefficients to each: \( 3(10 + (-3)) = 3(7) = 21 \). Choice D is incorrect because the student ignored the coefficient \( -2 \) on \( g \) entirely, computing \( 3(10) - 3 = 27 \).

Question 5. The function \( f(x) = x^2 - 4 \) is defined on \( [-3, 3] \). Which of the following correctly describes the relationship between \( \int_{-3}^{3} f(x)\,dx \) and \( \int_{-3}^{3} |f(x)|\,dx \)?

• A) They are equal because \( f \) is symmetric about the \( y \)-axis.
• B) \( \int_{-3}^{3} f(x)\,dx = 0 \) and \( \int_{-3}^{3} |f(x)|\,dx > 0 \)
• C) \( \int_{-3}^{3} f(x)\,dx \left|\int_{-3}^{3} f(x)\,dx\right| \) ✓
• D) \( \int_{-3}^{3} f(x)\,dx = \int_{-3}^{3} |f(x)|\,dx \) because \( f(x) \geq 0 \) on \( [-3, 3] \)

Explanation: Choice C is correct. Evaluating directly: \( \int_{-3}^{3}(x^2 - 4)\,dx = \left[\dfrac{x^3}{3} - 4x\right]_{-3}^{3} = (9 - 12) - (-9 + 12) = -3 - 3 = -6 \). The signed integral is negative because \( f(x) < 0 \) on \( (-2, 2) \), and the below-axis area outweighs the above-axis area. The integral of \( |f(x)| \) treats all area as positive; it adds the absolute value of the below-axis region rather than subtracting it, so \( \int_{-3}^{3}|f(x)|\,dx \) must be larger in magnitude than \( |-6| = 6 \). Choice A is incorrect because even-function symmetry means \( \int_{-3}^{3} f\,dx = 2\int_0^3 f\,dx \); it does not make the signed and unsigned integrals equal unless \( f \geq 0 \) everywhere on the interval. Choice B is incorrect because \( \int_{-3}^{3} f\,dx = 0 \) is a property of odd functions (\( f(-x) = -f(x) \)); since \( f(x) = x^2 - 4 \) is even, not odd, this is false. Choice D is incorrect because \( f(x) = x^2 - 4 < 0 \) on \( (-2, 2) \), so \( f \) is not non-negative on the entire interval.