If (a, 0) ; a>0 is the point where the curve y=sin2 x-√(3)...

If $(a, 0) ; a>0$ is the point where the curve $y=\sin 2 x-\sqrt{3} \sin x$ cuts the $X$ -axis first, $A$ is the area bounded by this part of the curve, the origin and the positive $X$ -axis, then
(a) $4 A+8 \cos a=7$
(b) $4 A+8 \sin a=7$
(c) $4 A-8 \sin a=7$
(d) $4 A-8 \cos a=7$

Key Concepts

Calculation of Areas Bounded by Curves

In calculus, finding the area bounded by a curve, the axes, or specific lines involves setting up definite integrals over the relevant interval. This process is key to understanding how to accumulate infinitesimal areas to determine the total area enclosed by given boundaries.

Integration of Trigonometric Functions

This concept pertains to evaluating the integral of functions that involve sine, cosine, and other trigonometric terms. Techniques such as substitution, using standard integral formulas, and leveraging trigonometric identities are commonly applied to compute the area under a curve defined by a trigonometric function.

Trigonometric Identities

This concept involves rewriting and simplifying expressions involving trigonometric functions by using known identities. For instance, using double angle formulas and factoring techniques helps in reducing complex expressions into simpler forms that are easier to analyze, particularly when determining zeros or simplifying integrals.

Finding Zeros of Trigonometric Functions (Intersection Points)

Identifying the points where a trigonometric function crosses the axis is essential in analysis and integration. It requires solving equations formed by setting the function equal to zero. This process typically involves breaking down the equation into simpler factors using identities, then solving for the variable within the proper range.

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