Question

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

\(y = \cos x,\quad y = x + 2{\sin ^4}x\)

Short answer

The area of the region is \(2.83259{\rm{ }}unit{s^2}\).

Step-by-step solution

The area \(A\) of the region bounded by the curves is given by

The area\(A\)of the region bounded by the curves\(y = f(x),y = g(x)\), and the lines\(x = a,x = b\), where\(f\)and\(g\)are continuous and\(f(x) \ge g(x)\)for all\(x\)in (a, b), is\(A = \int_a^b {(f(x) - g(x))dx} \)

Sketch the graph of the curves

Sketch the graph of the curve \(y = cos{\rm{ }}x\)and \(y = x + 2{\sin ^4}x\) as shown in the figure below

Calculate the area of the region enclosed

Evaluate the area bounded as follows

Based from Figure 1, there are two sub-regions that should be considered. The first region, which has upper bound functionand lower bound function, is on the interval. On the other hand, the second region, which has upper bound functionand lower bound function, is on the interval. Therefore, the area is given by:

Use a calculator to evaluate the integral, as the problem requires, the total area of the enclosed region is approximately: