find-the-area-between-the-function-y-2sin2x-1-and-the-x-axis-on-pi-pi-2-

Question Number 84135 by M±th+et£s last updated on 09/Mar/20

f i n d t h e a r e a b e t w e e n t h e f u n c t i o n

y = 2 s i n 2 x - 1 a n d t h e x - a x i s o n [- π, \frac{π}{2}]

Answered by Rio Michael last updated on 09/Mar/20

\int_{- π}^{\frac{π}{2}} (2 \sin 2 x - 1) d x

Commented by MJS last updated on 09/Mar/20

this is not the area

Answered by MJS last updated on 09/Mar/20

for the area we need the zeros

2 \sin 2 x - 1 = 0

\sin 2 x = \frac{1}{2} \land - π ⩽ x ⩽ \frac{π}{2}

\Rightarrow x \in {- \frac{11 π}{12}; - \frac{7 π}{12}; \frac{π}{12}; \frac{5 π}{12}}

y = 2 \sin (- 2 π) - 1 = - 1 < 0

\Rightarrow

area = - \underset{- π}{\int^{- \frac{11 π}{12}}} y d x + \underset{- \frac{11 π}{12}}{\int^{- \frac{7 π}{12}}} y d x - \underset{- \frac{7 π}{12}}{\int^{\frac{π}{12}}} y d x + \underset{\frac{π}{12}}{\int^{\frac{5 π}{12}}} y d x - \underset{\frac{5 π}{12}}{\int^{\frac{π}{2}}} y d x

\int y d x = \int (2 \sin 2 x - 1) d x = - x - \cos 2 x + C

\Rightarrow

area = \frac{π}{6} - 2 + 4 \sqrt{3} \approx 5.45180

Commented by M±th+et£s last updated on 10/Mar/20

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