What is area 2y=3sqrt(x),2y+3x=6,y=4 ?

The answer to area 2y=3sqrt(x),2y+3x=6,y=4 is 925/108

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area 2y=3sqrt(x),2y+3x=6,y=4

area

2 y = 3 x, 2 y + 3 x = 6, y = 4

\frac{925}{108}

Decimal

8.56481 \dots

Solution steps

Isolate

y

for

2 y = 3 x : y = \frac{3 x}{2}

Isolate

y

for

2 y + 3 x = 6 : y = \frac{6 - 3 x}{2}

f_{1} (x) = \frac{3 x}{2}

f_{2} (x) = \frac{6 - 3 x}{2}

To find the intersection points solve

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

\frac{3 x}{2} = \frac{6 - 3 x}{2} : x = 1

\frac{3 x}{2} = 4 : x = \frac{64}{9}

\frac{6 - 3 x}{2} = 4 : x = - \frac{2}{3}

= \int_{- \frac{2}{3}}^{1} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{1}^{\frac{64}{9}} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- \frac{2}{3}}^{1} \frac{6 - 3 x}{2} - 4 d x + \int_{1}^{\frac{64}{9}} \frac{3 x}{2} - 4 d x

\int_{- \frac{2}{3}}^{1} \frac{6 - 3 x}{2} - 4 d x = \frac{25}{12}

\int_{1}^{\frac{64}{9}} \frac{3 x}{2} - 4 d x = \frac{175}{27}

= \frac{25}{12} + \frac{175}{27}

Simplify

= \frac{925}{108}

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What is area 2y=3sqrt(x),2y+3x=6,y=4 ?
The answer to area 2y=3sqrt(x),2y+3x=6,y=4 is 925/108