What is area y=2x^2,y^2= 1/2 x ?

The answer to area y=2x^2,y^2= 1/2 x is 1/12

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area y=2x^2,y^2= 1/2 x

area

y = 2 x^{2}, y^{2} = \frac{1}{2} x

\frac{1}{12}

Decimal

0.08333 \dots

Solution steps

Isolate

y

for

y^{2} = \frac{1}{2} x : y = \frac{x}{2}, y = - \frac{x}{2}

f_{1} (x) = 2 x^{2}

f_{2} (x) = \frac{x}{2}

To find the intersection points solve

f_{i} (x) = f_{j} (x)

2 x^{2} = \frac{x}{2} : x = 0, x = \frac{1}{2}

2 x^{2} = - \frac{x}{2} : x = 0

\frac{x}{2} = - \frac{x}{2} : x = 0

= \int_{0}^{\frac{1}{2}} ∣ f_{1} (x) - f_{2} (x) ∣ d x

= \int_{0}^{\frac{1}{2}} 2 x^{2} - \frac{x}{2} d x

\int_{0}^{\frac{1}{2}} 2 x^{2} - \frac{x}{2} d x = \frac{1}{12}

= \frac{1}{12}