What is area x= 7/y ,x= y/9 ,x=9y ?

The answer to area x= 7/y ,x= y/9 ,x=9y is 28ln(3)

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area x= 7/y ,x= y/9 ,x=9y

area

x = \frac{7}{y}, x = \frac{y}{9}, x = 9 y

28 ln (3)

Decimal Notation

30.76114 \dots

Solution steps

Isolate

y

for

x = \frac{7}{y} : y = \frac{7}{x}; x \neq = 0

Isolate

y

for

x = \frac{y}{9} : y = 9 x

Isolate

y

for

x = 9 y : y = \frac{x}{9}

f_{1} (x) = \frac{7}{x}

f_{2} (x) = 9 x

To find the intersection points solve

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

\frac{7}{x} = 9 x : x = \frac{7}{3}, x = - \frac{7}{3}

\frac{7}{x} = \frac{x}{9} : x = 37, x = - 37

9 x = \frac{x}{9} : x = 0

= \int_{- 37}^{- \frac{7}{3}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{7}{3}}^{0} ∣ f_{3} (x) - f_{2} (x) ∣ d x + \int_{0}^{\frac{7}{3}} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{\frac{7}{3}}^{37} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 37}^{- \frac{7}{3}} \frac{7}{x} - \frac{x}{9} d x + \int_{- \frac{7}{3}}^{0} \frac{x}{9} - 9 x d x + \int_{0}^{\frac{7}{3}} 9 x - \frac{x}{9} d x + \int_{\frac{7}{3}}^{37} \frac{7}{x} - \frac{x}{9} d x

\int_{- 37}^{- \frac{7}{3}} \frac{7}{x} - \frac{x}{9} d x = 14 ln (3) - \frac{280}{81}

\int_{- \frac{7}{3}}^{0} \frac{x}{9} - 9 x d x = \frac{280}{81}

\int_{0}^{\frac{7}{3}} 9 x - \frac{x}{9} d x = \frac{280}{81}

\int_{\frac{7}{3}}^{37} \frac{7}{x} - \frac{x}{9} d x = 14 ln (3) - \frac{280}{81}

= 14 ln (3) - \frac{280}{81} + \frac{280}{81} + \frac{280}{81} + 14 ln (3) - \frac{280}{81}

Simplify

= 28 ln (3)

x \to 0 lim (2 x cot (x))

\frac{d}{d x} (\frac{( 2 x + x ^{2} )}{( 1 + x ) ^{2}})

\frac{d y}{d x} = y^{2} (1 + x^{2})

ln (3 - 2 x)

\int (\frac{x ^{2} + 9}{x ^{3}}) d x

What is area x= 7/y ,x= y/9 ,x=9y ?
The answer to area x= 7/y ,x= y/9 ,x=9y is 28ln(3)