Consider the function f(x)=sinx cosx. (a) Find the area...

Consider the function $f(x)=\sin x \cos x$.
(a) Find the area between the graphs of $f(x)$ and $g(x)=$ $\sin x$ on $[0, \pi]$ shown next at the left.
(b) Find the area between the graphs of $f(x)$ and $h(x)=$ $\sin 2 x$ on $[0, \pi]$ shown next at the right.

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Key Concepts

Integration Techniques

The computation of areas between curves often involves techniques of integration, such as applying substitution, using known antiderivatives, or taking advantage of symmetry properties of the functions. Mastery of these techniques is essential to evaluate the definite integrals that represent the area.

Trigonometric Identities

This concept involves recognizing and using fundamental trigonometric relationships, such as the double angle identity which expresses sin x cos x as (1/2) sin 2x. This identity can simplify the expressions involved, making integrals and comparisons easier to handle.

Area Between Curves

The area between two curves is found by integrating the absolute difference of the functions over a specified interval. This method requires determining which function is upper and which is lower over subintervals, and then computing separate integrals if necessary to capture the entire area between the curves.

Intersection Points

Identifying the intersection points of the curves is crucial for setting up the correct limits of integration. By solving the equation f(x) = g(x) (or f(x) = h(x)), one determines the x-values where the curves meet, which then partition the interval into regions where one function is above the other.

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