Question

Find the area of the region. Use a graphing utility to verify your result. $$ y=2 \sin x+\sin 2 x $$

Short answer

The area under the curve \(y = 2\sin(x) + \sin(2x)\) from \(0\) to \(2\pi\) is equal to \(0\).

Step-by-step solution

Identify the Integral Boundaries

The integral boundaries are determined by the period of the function, which is here \(2\pi\). So the boundaries will range from 0 to \(2\pi\).

Set Up the Integral

The area under the curve is found by integrating: \(\int_a^b f(x) \,dx\). Set up the integral for given function \(y = 2\sin(x) + \sin(2x)\) from 0 to \(2\pi\). So, the equation becomes: \(\int_0^{2\pi} [2\sin(x) + \sin(2x)] \,dx\).

Evaluate the Integral

Evaluate the integral. This involves integrating each of the parts separately. The antiderivative of \(2\sin(x)\) is \(-2\cos(x)\) and the antiderivative of \(\sin(2x)\) is \(-\frac{1}{2}\cos(2x)\). After evaluating these at \(2\pi\) and subtracting the value at \(0\), get the final result.