Question

Find the area of the region enclosed by the curves. 2 y=x²-81 and y=x^2/2+81​

Answer 1

The total area of the regions between the curves is 3888 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 = x² - 81 and y = x²/2 + 81

With the use of graphs, the curves intersect at

x = -18 and x = 18

This represents the limit of the integration

So, the area of the regions between the curves is

`\text{Area} = \int\limits^(18)_(-18) [x\² - 81 - (x\²)/(2) - 81] dx`

This gives

`\text{Area} = \int\limits^(18)_(-18) [\frac12x\² - 162] dx`

Integrate

`\text{Area} = [\frac16x\³ - 162x]|\limits^(18)_(-18)`

Expand

Area = [1/6(-18)³ - 162(-18)] - [1/6(18)³ - 162(18)]

Evaluate

Area = 3888

Hence, the total area of the regions between the curves is 3888 square units