AP Calculus BC 6.7: FTC Definite Integrals

Practice AP Calculus BC questions on evaluating definite integrals analytically by applying antiderivatives and 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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FUN-6.B—Evaluate definite integrals analytically using the Fundamental Theorem of Calculus question 1