Final answer:
To determine the area of the region bounded by y=x¹/², y= 1/2 x, and x=9, we split it into two sections and calculate each area separately, then sum them. Upon calculation, the total area was found to be approximately 18.92 square units, which is not an option provided in the question.
Step-by-step explanation:
To find the area of the region bounded by the curves y=x¹/², y= 1/2 x, and x=9, we first identify the points of intersection between the curves y=x¹/² and y=1/2 x. The intersection occurs when x¹/² = 1/2 x, which simplifies to x¹/² = x/2, or 2x¹/² = x. Solving this gives us x=0 and x=4 (since the square root function cannot be negative). The bounded region is therefore from x=0 to x=4 for the curve y=x¹/², and from x=4 to x=9 for the line y=1/2 x.
To calculate the area, we split it into two parts: the area under y=x¹/² from x=0 to x=4, and the area under y=1/2 x from x=4 to x=9. The total area is the sum of these two areas.
Area under y=x¹/² from x=0 to x=4 is given by the integral of x¹/² dx from 0 to 4, which equals ⅓x³/² from 0 to 4, equal to ⅓(4³/² - 0) = ⅓× 8 = 8/3 or approximately 2.67 square units.
Area under y=1/2 x from x=4 to x=9 is a trapezoid with vertices (4,2), (9,4.5), (9,0), and (4,0). The area of a trapezoid is the average of the bases times the height, which in this case is ((2+4.5)/2)×(9-4) = (6.5/2)× 5 = 16.25 square units.
The total area is the sum of both areas, which is approximately 2.67 + 16.25 = 18.92 square units. However, this value is not listed as an option in the question, suggesting there may have been an error in interpreting the boundaries or in the solution process. Please verify that the provided boundaries and equations of lines/curves are as intended before calculating the area.