Find the exact area of the region bounded by the graphs with equations:y = x + 3 and y = 12 + x − x2

To find the exact area of the region bounded by the graphs with equations y = x + 3 and y = 12 + x - x², we need to follow these steps:

Step 1: Set the equations equal to each other to find the points of intersection.

x + 3 = 12 + x - x²

Rearrange the equation to isolate terms:

x² - x + 3 - 12 = 0

Simplify to:

x² - x - 9 = 0

Factor the equation:

(x - 3)(x + 3) = 0

Setting each factor equal to zero gives the solutions x = 3 and x = -3. These are the x-values where the two graphs intersect.

Step 2: Sketch the graphs to determine which function is the upper function and which is the lower function in the interval [-3, 3]. You will find that y = x + 3 is the lower function and y = 12 + x - x² is the upper function.

Step 3: Set up the integral to find the area between the curves. The formula for the area between two curves is ∫ from a to b [f(x) - g(x)] dx, where f(x) is the upper function and g(x) is the lower function.

So, the integral becomes:

∫ from -3 to 3 [(12 + x - x²) - (x + 3)] dx

Simplify the integrand:

∫ from -3 to 3 [9 - x²] dx

Step 4: Evaluate the integral.

[9x - (x³/3)] from -3 to 3

= [27 - 9] - [(-27 - 9)] = 18 - (-36) = 54

So, the exact area of the region bounded by the graphs with equations y = x + 3 and y = 12 + x - x² is 54 square units.