AP Calculus BC 6.7 FTC Definite Integrals Overview

Review definite integrals by selecting antiderivatives and applying the Fundamental Theorem with correct bounds, signs and evaluation at both endpoints.

Covered syllabus points

FUN-6.B—Evaluate definite integrals analytically using the Fundamental Theorem of Calculus

[Maximum number: 2]

The continuous function f is defined on the closed interval $-6 \leq x \leq 5$ . The figure above shows a portion of the graph of f, consisting of two line segments and a quarter of a circle centered at the point (5, 3). It is known that the point $(3,3-\sqrt{5})$ is on the graph of f.

Evaluate $\int_{3}^{5}\left(2 f^{\prime}(x)+4\right) d x$ .

\begin{aligned} \int_{3}^{5}\left(2 f^{\prime}(x)+4\right) d x & =2 \int_{3}^{5} f^{\prime}(x) d x+\int_{3}^{5} 4 d x \\ & =2(f(5)-f(3))+4(5-3) \\ & =2(0-(3-\sqrt{5}))+8 \\ & =2(-3+\sqrt{5})+8=2+2 \sqrt{5} \end{aligned}

\text { - } \mathrm{OR}-

\begin{aligned} \int_{3}^{5}\left(2 f^{\prime}(x)+4\right) d x & =[2 f(x)+4 x]_{x=3}^{x=5} \\ & =(2 f(5)+20)-(2 f(3)+12) \\ & =(2 \cdot 0+20)-(2(3-\sqrt{5})+12) \\ & =2+2 \sqrt{5} \end{aligned}

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Covered syllabus points

6.7 The Fundamental Theorem of Calculus and Definite Integrals question 1