Final answer:
To find the area of the region bounded by the functions f(x) and g(x), we need to find the points of intersection between the two curves. We then integrate the difference between the two functions from the lower x-value to the upper x-value to find the area under the curve. The area of the region R is approximately 69.855 square units.
Step-by-step explanation:
To find the area A of the region R bounded by the functions f(x) and g(x), we need to find the points where f(x) and g(x) intersect. We can do this by setting the two functions equal to each other and solving for x: (x^(2))/(2)-(5x)/(2)-2 = -(3x^(2))/(4)-(5x)/(2)+3. Simplifying this equation gives us 7x^(2) - 7x - 16 = 0. Solving this quadratic equation, we find x = 4 and x = -0.571. These are the x-values where the two functions intersect.
To find the y-values corresponding to these x-values, we substitute them back into one of the functions. Using f(x), we find that f(4) = 4 and f(-0.571) = -4.142. Now we have the points of intersection: (4, 4) and (-0.571, -4.142).
Next, we integrate the difference between the two functions from x = -0.571 to x = 4 to find the area under the curve. The formula for the area between two curves is A = ∫(g(x) - f(x)) dx. Evaluating this integral gives us the area A = ∫(-((3x^(2))/4) - ((5x)/2) + 3 - (x^(2))/2 + (5x)/2 + 2) dx = ∫(-5x^(2)/4 + 5) dx. Integrating this expression gives us A = [-5x^(3)/(12)] + [5x] evaluated from -0.571 to 4. Evaluating this expression gives us A ≈ 69.855 square units.