CPM Homework Help : CALC3RD Problem 9-111

Calculate the area of the region which includes the origin and is bounded by the curves $f(x) = x + 1$ , $g(x) = x^2 - 1$ , and $h(x) = \cos^{–1}(x)$ .

Step 1:

Graph the curves and determine the points of intersection.

Upward parabola with vertex at the point (0, comma negative 1), decreasing concave down curve coming from upper left, ending at the point (1, comma 0), which is on the parabola, & an increasing line intersecting the parabola at the point (negative 1, comma 0), & intersecting the decreasing curve at the point (0.283, comma 1.283).

Step 2:

$\int_{-1}^{0.283}\big(f(x)-g(x)\big)dx+\int_{0.283}^1\big(h(x)-g(x)\big)dx$

More Help:

For the second integral, use integration by parts.
Let $f = \cos^{–1}(x)$ and $dg = 1dx$ .