Find the area of the region bounded by the graphs of y1x and

Solution

sol Determine the x-coordinates of two intersection points for both equations by substitution method: 2x + 2(1/x) = 5 x[2x + 2/x = 5] 2x² + 2 = 5x 2x² - 5x + 2 = 0 (2x - 1)(x - 2) = 0 2x - 1 = 0 and x - 2 = 0 x = ½, 2 Note that 2x + 2y = 5 represents the upper function while y = 1/x represents the lower function. Before setting up the integral with x limit, you need to solve for y for 2x + 2y = 5. 2y = -2x + 5 y = -x + 5/2 Then we get A = ?½ to 2 upper - lower dx ==> ?½ to 2 (-x + 5/2) - 1/x dx ==> ?½ to 2 -x + 5/2 - 1/x dx ==> -x²/2 + 5x/2 - ln(x) | ½ to 2 ftc using ==> -(2)²/2 + 5(2)/2 - ln(2) - (-(½)²/2 + 5(½)/2 - ln(½)) ==> -2 + 5 - ln(2) - (-(¼)/2 + 5/4 - ln(½)) ==> 3 - ln(2) - (-1/8 + 5/4 - ln(½)) ==> 3 - ln(2) - (9/8 - ln(½)) ==> 15/8 - (ln(2) - ln(½)) ==> 15/8 - ln(2/½) ==> 15/8 - ln(4) ans