Question

Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area. give your answer correct to 3 decimal places.

Answer 1

b = ∛16 ≈ 2.520

Explanation:

The area bounded by y=x^2 and y=b is given by the integral ...

`\displaystyle A(b)=\int\limits^(√(b))_(-√(b)) {(b-x^2)} \, dx=2b√(b)-(2)/(3)b√(b)=(4)/(3)b^{(3)/(2)}`

For b = 4,

A(4) = (4/3)4^(3/2) = 32/3

We want to find b such that the area is half that, or ...

A(b) = (1/2)(32/3) = 16/3

So, we're solving the equation ...

16/3 = 4/3b^(3/2)

4 = b^(3/2) . . . . . . . . . . . . . . multiply by 3/4

4^(2/3) = b = ∛16 . . . . . . . . raise to the 3/2 power