What is area y=x,y=5x,y=-x+2 ?

The answer to area y=x,y=5x,y=-x+2 is 2/3

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area y=x,y=5x,y=-x+2

area

y = x, y = 5 x, y = - x + 2

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

To find the intersection points solve

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

x = 5 x : x = 0

x = - x + 2 : x = 1

5 x = - x + 2 : x = \frac{1}{3}

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

= \int_{0}^{\frac{1}{3}} ∣ x - 5 x ∣ d x + \int_{\frac{1}{3}}^{1} ∣ x - (- x + 2) ∣ d x

\int_{0}^{\frac{1}{3}} ∣ x - 5 x ∣ d x = \frac{2}{9}

\int_{\frac{1}{3}}^{1} ∣ x - (- x + 2) ∣ d x = \frac{4}{9}

= \frac{2}{9} + \frac{4}{9}

Simplify

= \frac{2}{3}

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