What is area y=x-1,y^2=2x+6 ?

The answer to area y=x-1,y^2=2x+6 is 18

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area y=x-1,y^2=2x+6

area

y = x - 1, y^{2} = 2 x + 6

18

Solution steps

Isolate

y

for

y^{2} = 2 x + 6 : y = 2 x + 6, y = - 2 x + 6

f_{1} (x) = x - 1

f_{2} (x) = 2 x + 6

To find the intersection points solve

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

x - 1 = 2 x + 6 : x = 5

x - 1 = - 2 x + 6 : x = - 1

2 x + 6 = - 2 x + 6 : x = - 3

= \int_{- 3}^{- 1} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{- 1}^{5} ∣ f_{1} (x) - f_{2} (x) ∣ d x

= \int_{- 3}^{- 1} 2 x + 6 - (- 2 x + 6) d x + \int_{- 1}^{5} x - 1 - 2 x + 6 d x

\int_{- 3}^{- 1} 2 x + 6 - (- 2 x + 6) d x = \frac{16}{3}

\int_{- 1}^{5} x - 1 - 2 x + 6 d x = \frac{38}{3}

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

Simplify

= 18

\int 4 e^{x} 10 + e^{x} d x

\frac{1 - 5 t}{6 + t}

\frac{\partial}{\partial x} (x^{4} - a x^{2} + a^{2})

n = 1 \sum \infty n^{\frac{3}{2}}

t \cdot \frac{d y}{d t} = t^{3} + 11 t^{3} y