substitution 21.8

Question

The curve to the right is the graph of the equation

y=−x16−x2.

Find the total area of the shaded regions in the graph.

Answer 1

Since a shaded region could not be supplied, this solution assumes that the shaded region is that bounded by the curves y = x² and y = x¹⁶.

First, consider 0 < x < 1. If p < q, then x^p > x^q. Also, since both curves pass through (0, 0) and (1,1), a small region is bounded by the two curves, with y = x² "above" the curve y = x¹⁶.

So the area of this region is given by the integral:

S x² - x¹⁶ dx (using 0 and 1 as the lower and upper bounds of the integral)

The integration result is 1/3 x³ - 1/17 x¹⁷ evaluated at 1 and 0

= 0.3333 (1) - 0.0588 (1) - [0 - 0] = 0.2745

Finally, since the integrand is an even function, the above discussion "consider 0 < x < 1" also applies to

-1 < x < 0, producing another shaded region of identical size.

FINAL ANSWER: 2( 0.2745) = 0 5490