Find the area of the region bounded by the curves y = x2 − 1 and y = cos(x). Give your answer correct to 2 decimal places.

Question

Find the area of the region bounded by the curves y = x² − 1 and y = cos(x). Give your answer correct to 2 decimal places.

Answer 1

The first thing to do is find the intersection points of the two functions because those will be the limits of integration. That's not algebraically simple so the easiest thing to do is to graph the two functions and find the intersection points using your graphing calculator. Mine gives my x = ± 1.176502

So the area of the region bounded is ∫cos(x)dx - ∫(x² - 1)dx using those ± 1.176502 limits of integration.

Plugging those into my calculator, I get the area is 3.1139 so to 2 decimal places it is 3.11 square units.

If you want to do it manually, you can take antiderivatives and plug in the limits of integration but I can't see much of a point in doing so.

Answer 2

You have to determine where y=x^2-1 and y=cos(x) intersect. Approximately x=-1.177 and x=1.177

At x=0, cos(x)=1 and x^2-1=-1 so cos(x) is above x^2-1

Area = ∫ cos(x) - (x^2-1) dx (integration limits -1.177 and 1.177)