What is area x+y^2=30,x+y=0 ?

The answer to area x+y^2=30,x+y=0 is 1331/6

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area x+y^2=30,x+y=0

area

x + y^{2} = 30, x + y = 0

\frac{1331}{6}

Decimal Notation

221.83333 \dots

Solution steps

Isolate

y

for

x + y^{2} = 30 : y = 30 - x, y = - 30 - x

Isolate

y

for

x + y = 0 : y = - x

f_{1} (x) = 30 - x

f_{2} (x) = - 30 - x

To find the intersection points solve

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

30 - x = - 30 - x : x = 30

30 - x = - x : x = - 6

- 30 - x = - x : x = 5

= \int_{- 6}^{5} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{5}^{30} ∣ f_{2} (x) - f_{1} (x) ∣ d x

= \int_{- 6}^{5} 30 - x - (- x) d x + \int_{5}^{30} - 30 - x - 30 - x d x

\int_{- 6}^{5} 30 - x - (- x) d x = \frac{331}{6}

\int_{5}^{30} - 30 - x - 30 - x d x = \frac{500}{3}

= \frac{331}{6} + \frac{500}{3}

Simplify

= \frac{1331}{6}

\int \frac{y ^{5}}{( 1 - y ^{3} ) ^{\frac{3}{2}}} d y

f (x) = (x + x^{4}) (- x - 1)

y^{^{'}} sin (2 π x) = π y cos (2 π x)

x \to 1 - lim (4 - x - x^{2})