Question

Sketch the region bounded by the curves y = -(1/9)x² + 6, y = √(x-2), y = 2-x, and the y-axis, and find its area.

Answer 1

The total area of the regions between the curves is 13.83 square units

Calculating the total area of the regions between the curves

From the question, we have the following parameters that can be used in our computation:

y = -(1/9)x² + 6, y = √(x-2), y = 2-x,

With the use of graphs, the curves intersect at

x = -3 and x = 6

So, the area of the regions between the curves is

Area = ∫-(1/9)x² + 6 - √(x-2) - 2 + x

This gives

Area = ∫-(1/9)x² + 4 - √(x-2) + x

Integrate

`Area = -(x^3)/(27)+(x^2)/(2)+4x-(2\left(x-2\right)^(3)/(2))/(3)`

Using the limits, we have

So, we have

Area = 83/6

Evaluate

Area = 13.83

Hence, the total area of the regions between the curves is 13.83 square units