What is area 2x+y^2=8,x=y ?

The answer to area 2x+y^2=8,x=y is 18

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area 2x+y^2=8,x=y

area

2 x + y^{2} = 8, x = y

18

Solution steps

Isolate

y

for

2 x + y^{2} = 8 : y = 8 - 2 x, y = - 8 - 2 x

Isolate

y

for

x = y : y = x

f_{1} (x) = 8 - 2 x

f_{2} (x) = - 8 - 2 x

To find the intersection points solve

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

8 - 2 x = - 8 - 2 x : x = 4

8 - 2 x = x : x = 2

- 8 - 2 x = x : x = - 4

= \int_{- 4}^{2} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{2}^{4} ∣ f_{1} (x) - f_{2} (x) ∣ d x

= \int_{- 4}^{2} - 8 - 2 x - x d x + \int_{2}^{4} 8 - 2 x - (- 8 - 2 x) d x

\int_{- 4}^{2} - 8 - 2 x - x d x = \frac{38}{3}

\int_{2}^{4} 8 - 2 x - (- 8 - 2 x) d x = \frac{16}{3}

= \frac{38}{3} + \frac{16}{3}

Simplify

= 18

\frac{\partial}{\partial x} (6 x y - 2 y^{2})

\frac{d}{d x} (\frac{4}{7 x ^{2}})

\int \frac{x + 7}{x ^{2} + 3 x + 2} d x

y = \frac{1}{x - 2}

\frac{d y}{d x}, e^{(9 x y)} = 10 x