AP Calculus AB 6.6 Applying Properties of Definite Integrals - MCQs - Exam Style Questions
No-Calc Question
The function \(f\) is defined for \(-2\le x\le 5\), and the graph of \(y=f(x)\) shown consists of three line segments. What is the value of \(\displaystyle\int_{-2}^{5}\big(f(x)+1\big)\,dx\) ?
(A) \(-10\)
(B) \(-9\)
(C) \(-3\)
(D) \(17\)
▶️ Answer/Explanation
Split the integral: \(\displaystyle\int_{-2}^{5}f(x)\,dx+\int_{-2}^{5}1\,dx\).
From areas of the three line segments in the graph, \(\displaystyle\int_{-2}^{5}f(x)\,dx=-10\).
Also \(\displaystyle\int_{-2}^{5}1\,dx=5-(-2)=7\).
Sum: \(-10+7=-3\).
✅ Answer: (C)
No-Calc Question
If \(\displaystyle\int_{1}^{4} f(x)\,dx=8\) and \(\displaystyle\int_{1}^{4} g(x)\,dx=-2\), which of the following cannot be determined from the information given?
(A) \(\displaystyle\int_{4}^{1} g(x)\,dx\)
(B) \(\displaystyle\int_{1}^{4} 3f(x)\,dx\)
(C) \(\displaystyle\int_{1}^{4} 3f(x)g(x)\,dx\)
(D) \(\displaystyle\int_{1}^{4} \big(3f(x)+g(x)\big)\,dx\)
(B) \(\displaystyle\int_{1}^{4} 3f(x)\,dx\)
(C) \(\displaystyle\int_{1}^{4} 3f(x)g(x)\,dx\)
(D) \(\displaystyle\int_{1}^{4} \big(3f(x)+g(x)\big)\,dx\)
▶️ Answer/Explanation
(A): Reverse limits ⇒ \(\int_{4}^{1}g=-\int_{1}^{4}g=2\) (known).
(B): Constant multiple ⇒ \(3\int_{1}^{4}f=24\) (known).
(D): Linearity ⇒ \(3\int f+\int g=24-2=22\) (known).
(C): Product integral has no given value — cannot be deduced.
✅ Answer: (C)