Calculus: Early Transcendentals, Chapter 6, 6.1, Section 6.1, Problem 33

The red curve refers to the graph of the first function: y =xsin(x^2) while the the the blue curve refers to the graph of the second function: y=x^4 .



As shown in the xy-plane, the two graphs intersect , approximately, at the following points: (0,0) and (0.9,0.65).
Based on these intersection points, the limits of integration with respect to x will be from x= 0 to x=0.9.
The formula for the " Area between Two Curves" is:
A= int_a^b[f(x)-g(x)]dx
such that f(x)gt=g(x) on the interval of [a,b].
This is the same as A = int_a^b[y_(above) - y_(below)]dx
where the bounded area is in between y_(above) = f(x) and y_(below)= g(x) .

Applying the formula on the given problem, the integration will be:
A = int_0^(0.9)[x*sin(x^2) - x^4]dx
= [-(cos(x^2)/2) -x^5/5] |_0^(0.9)
= [-(cos((0.9)^2)/2) -(0.9)^5/5]-[-(cos((0)^2)/2) -0^5/5]
=-0.4628472164 - (-0.5)
= -0.4628472164 + 0.5
= 0.03715278360
~~ 0.0372 as the Area of the region bounded by the curves shown above.