Approximate x-coordinates of the points of intersection of the given curves, and find the area of the region bounded by the curves.
i) The intersection points of the curves:
x=0 and x≈0.896
ii) The area of the region bounded by the curves:
The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b is
fx-gx=fx-gx when fx≥g(x)gx-fx when gx≥f(x)
By substitution u=x2∫xsinx2= -12cosx2+c
y=xsinx2, y=x4, x≥0
fx=xsinx2, and gx=x4
i) To find the intersection point of the curves, draw the graph.
From the sketch, it is clear that the curves intersect at x=0 and x=0.896
ii) the upper curve is y=x sinx(x2) and the lower curve is y=x4.