What is area x+y^2=42,x=y ?

The answer to area x+y^2=42,x=y is 2197/6

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area x+y^2=42,x=y

area

x + y^{2} = 42, x = y

\frac{2197}{6}

Decimal Notation

366.16666 \dots

Solution steps

Isolate

y

for

x + y^{2} = 42 : y = 42 - x, y = - 42 - x

Isolate

y

for

x = y : y = x

f_{1} (x) = 42 - x

f_{2} (x) = - 42 - x

To find the intersection points solve

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

42 - x = - 42 - x : x = 42

42 - x = x : x = 6

- 42 - x = x : x = - 7

= \int_{- 7}^{6} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{6}^{42} ∣ f_{1} (x) - f_{2} (x) ∣ d x

= \int_{- 7}^{6} - 42 - x - x d x + \int_{6}^{42} 42 - x - (- 42 - x) d x

\int_{- 7}^{6} - 42 - x - x d x = \frac{469}{6}

\int_{6}^{42} 42 - x - (- 42 - x) d x = 288

= \frac{469}{6} + 288

Simplify

= \frac{2197}{6}

f (x) = x - 2, at x = 6

\frac{d ^{2}}{d x ^{2}} (\frac{8}{9 t ^{5}})

y = 4 cos (x), y = 6 - 2 cos (x), 0 \leq x \leq 2 π

\frac{d y}{d x} = \frac{cos ( 8 x )}{e ^{8 y}}, y (0) = 1

f (x) = x, at x = 49