First slide

Evaluation of definite integrals

Question

$Evaluate \int_{0}^{16 π / 3} | \sin x | d x$

Moderate

A

$\frac{21}{2}$
B

0
C

$\frac{19}{2}$
D

$\frac{17}{2}$

Solution

$\begin{array}{l} \begin{matrix} \int_{0}^{16 π / 3} | \sin x | d x & = \int_{0}^{5 π} | \sin x | d x + \int_{5 π}^{5 π + π / 3} | \sin x | d x \\ = 5 \int_{0}^{π} | \sin x | d x + \int_{0}^{π / 3} | \sin x | d x \end{matrix} \\ [∵ | \sin x | is periodic with period π \\ = 5 \int_{0}^{π} \sin x d x + \int_{0}^{π / 3} \sin x d x \\ \begin{aligned} = 5 [- \cos x]_{0}^{π} + [- \cos x]_{0}^{\frac{π}{3}} \\ = 5 \times 2 + (- \frac{1}{2} + 1) = \frac{21}{2} \end{aligned} \end{array}$

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