In each of the following problrmes, our goal is to determine the area of the region described. for each region, (i) determine the intersection points of the curves, (ii) sketch the region whose area is being found, (iii) draw and label a representative slice, and (iv) state the area of the representative slice. then, state a definite integral whose value is the exact area of the region, and evaluate the integral to find the numeric value of the region's area.
(a) the finite region bounded by y = √(x) and y = (1/4)x.
(b) the finite region bounded by y = 12 - 2x^2 and y = (x^2) - 8.
(c) the area bounded by the y-axis, f(x) = cos(x), and g(x) = sin(x), where we consider the region formed by the first positive value of x for which f and g intersect.
(d) the finite regions between the curves y = (x^3) - x and y = x^2.