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.

The accompanying table shows the numbers of male and female students in a particular country who received bachelor's degrees in business in a recent year. complete parts (a) and (b) below. business degrees nonbusiness degrees total male 189131 634650 823781 female 169539 885329 1054868 total 358670 1519979 1878649 (a) find the probability that a randomly selected student is male, given that the student received a business degree. the probability that a randomly selected student is male, given that the student received a business degree, is nothing. (round to three decimal places as needed.) (b) find the probability that a randomly selected student received a business degree, given that the student is female. the probability that a randomly selected student received a business degree, given that the student is female, is nothing. (round to three decimal places as needed.)

The avenues in a particular city run north to south and are numbered consecutively with 1st avenue at the western border of the city. the streets in the city run east to west and are numbered consecutively with 1st street at the southern border of the city. for a festival, the city is not allowing cars to park in a rectangular region bordered by 5th avenue to the west. 9th avenue to the east, 4th street to the south, and 6th street to the north. if x is the avenue number and yis the street number, which of the following systems describes the region in which cars are not allowed to park? 5th ave 9th ave

The graph of y x3 is translated so that the point (1. 1) is moved to (1, 4). what is the equation of the new graph?