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Geometric Interpretation of Def.Int as Limit of Sum

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Question

Evaluate: $\int_{π / 6}^{π / 4} cos 2 x d x =$

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Solution

Consider, $I = \int_{\frac{π}{6}}^{\frac{π}{4}} cos 2 x$

$= {[sin 2 x]}_{\frac{π}{6}}^{\frac{π}{4}}$

$= sin \frac{π}{2} - sin \frac{π}{3} = 1 - \frac{\sqrt{3}}{2}$

$= \frac{2 - \sqrt{3}}{2}$

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\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{x + \frac{π}{4}}{2 - c o s 2 x} d x .

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Geometric Interpretation of Def.Int as Limit of Sum

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