Question

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=x^{2}, \quad y=4 \cos x $$

Short answer

The computational facilities of a graphing utility can calculate the approximate area between the two functions \(y = x^{2}\) and \(y = 4cos(x)\) with ease and good accuracy because of their integration functions.

Step-by-step solution

Graph the Functions

Start by using a graphing utility to plot the functions \(y = x^{2}\) and \(y = 4cos(x)\). Observe where they intersect.

Analyze the Difficulty of Manual Calculation

The area of the region between these two functions would require to calculate two separate definite integrals, for the areas on the left and on the right of the intersection points. In addition, the integral of \(cosx\) is not a simple function, making the calculation difficult by hand.

Approximate the Area Using Graphing Utility

Use the integration feature of the graphing utility to calculate the area. Use definite integrals from the intersection points of the two functions and sum them up, allowing the program to solve for the integral and approximate the area to four decimal places.