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

Answer:

  b = ∛16 ≈ 2.520

Step-by-step explanation:

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

$\displaystyle A(b)=\int\limits^{\sqrt{b}}_{-\sqrt{b}} {(b-x^2)} \, dx=2b\sqrt{b}-\frac{2}{3}b\sqrt{b}=\frac{4}{3}b^{\frac{3}{2}}$

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