Rectangle areas are found by calculating height × width. the width of each rectangle equals δx and the height of each rectangle is given by the function value at the right-hand side of the rectangle. so we must calculate r4 = 4 f(xi)δx = [f(x1) + f(x2) + f(x3) + f(x4)] δx i = 1 , where x1, x2, x3, x4 represent the right-hand endpoints of four equal sub-intervals of 0, π 2 . since we wish to estimate the area over the interval 0, π 2 using 4 rectangles of equal widths, then each rectangle will have width δx = .

Ahigh-altitude spherical weather balloon expands as it rises, due to the drop in atmospheric pressure. suppose that the radius r increases at the rate of 0.02 inches per second, and that r = 36 inches at time t = 0. determine the equation that models the volume v of the balloon at time t, and find the volume when t = 360 seconds. v(t) = 4π(0.02t)2; 651.44 in3 v(t) = 4π(36 + 0.02t)2; 1,694,397.14 in3 v(t) = four pi times the product of zero point zero two and t to the third power divided by three.; 4,690.37 in3 v(t) = four pi times the quantity of thirty six plus zero point zero two t to the third power divided by three.; 337,706.83 in3