Question

The area bounded by curve y=sin2x, x-axis and the lines x=π/4 and 3π/4 is​

Answer 1

We have to find the area bounded by curve y = sin2x , x - axis and lines x = π/4 and 3π/4.

solution : see the diagram as shown in figure. here it is clear that area bounded by curve above the x - axis is same as that of below the x - axis.

so, we have to find area between two interval.

π/4 to π/2 and π/2 to 3π/4 and then we have to add both areas.

now area bounded by the curve = $\left|\int\limits^{π/4}_{\pi/2}{sin2x}\,dx\right|+\left\int\limits^{\pi/2}_{3\pi/4}{sin2x}\,dx\right|$

= $\left|\left[\frac{-cos2x}{2}\right]^{\pi/4}_{\pi/2}\right|+\left|\left[\frac{-cos2x}{2}\right]^{\pi/2}_{3\pi/4}\right|$

= |-1/2(cosπ/2 - cosπ)| + |-1/2(cosπ - cos3π/2)|

= |-1/2 × 1 | + |-1/2 × -1|

= 1/2 + 1/2

= 1

Therefore the area bounded by the curves is 1 unit