Find the area of the shaded region for the graph given below.

Question

Find the area of the shaded region for the graph given below. ​  ​

Quizplus · Accepted Answer

To find the shaded region, we need to find the area between the curve and the x-axis from x = 0 to x = 3. First, we need to find the points of intersection.
Set the two equations equal to each other:
x^2 = 9 - x
x^2 + x - 9 = 0
Using the quadratic formula, we get x = -1 - √37/2 and x = -1 + √37/2. Only the positive value, x = -1 + √37/2 ≈ 1.06, is within the given domain of x = [0,3].

Next, we integrate the curve from x = 0 to x = 1.06:
∫[0,1.06] (x - x^2 - 3) dx = [x^2/2 - x^3/3 - 3x] [0,1.06] = 1.25

Thus, the area of the shaded region is approximately 1.25 units^2.

The only answer choice that matches this value is C.

Find the Area of the Shaded Region for the Graph