AP Calculus AB Unit 6.6: Integral Properties

Practice AP Calculus Unit 6.6 questions on evaluating definite integrals from geometry and applying linearity, interval, and orientation properties.

Exam points

evaluate definite integrals using geometric areas and properties such as additivity, linearity, and reversed limits

Covered syllabus points

FUN-6.A—Calculate a definite integral using areas and properties of definite integrals

[Maximum number: 2]

The continuous function f is defined on the closed interval $-6 \leq x \leq 12$ . The graph of f,

consisting of two semicircles and one line segment, is shown in the figure.

$Graph of $f$$

Graph of $f$

Let g be the function defined by $g(x)=\int_{6}^{x} f(t) d t$ .

Find g(12) and g(0). Label your answers.

C Find g(12) and g(0). Label your answers.

$g(12)=\int_{6}^{12} f(t) d t=\frac{1}{2} \cdot 6 \cdot 3=9$	g(12)	Point 5 (P5)
$g(0)=\int_{6}^{0} f(x) d x=-\int_{0}^{6} f(x) d x=-\frac{\pi}{2} 3^{2}=-\frac{9 \pi}{2}$	g(0)	Point 6 (P6)

Scoring Notes for Part C

- Unlabeled values do not earn either P5 or P6.

- P5 is earned for a response of g(12)=9, with or without supporting work.

- P6 is earned for a response of $g(0)=-\frac{9 \pi}{2}$ , with or without supporting work.

Note: Incorrect communication between the label " g(0) " and the answer will be treated as scratch

work and will not impact scoring. For example, $g(0)=\int_{0}^{6} f(x) d x=-\frac{9 \pi}{2}$ earns P6.

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Covered syllabus points

6.6 Applying Properties of Definite Integrals question 1