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Popular Calculus >

area x=y,x=-y^2+6y

Solution

area

x = y, x = - y^{2} + 6 y

Solution

\frac{125}{6}

+1

Decimal

20.83333 \dots

Solution steps

Isolate

y

for

x = y : y = x

Isolate

y

for

x = - y^{2} + 6 y : y = 3 - - x + 9, y = - x + 9 + 3

f_{1} (x) = x

f_{2} (x) = 3 - - x + 9

To find the intersection points solve

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

x = 3 - - x + 9 : x = 0

x = - x + 9 + 3 : x = 5

3 - - x + 9 = - x + 9 + 3 : x = 9

= \int_{0}^{5} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{5}^{9} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{0}^{5} x - (3 - - x + 9) d x + \int_{5}^{9} 3 - - x + 9 - (- x + 9 + 3) d x

\int_{0}^{5} x - (3 - - x + 9) d x = \frac{61}{6}

\int_{5}^{9} 3 - - x + 9 - (- x + 9 + 3) d x = \frac{32}{3}

= \frac{61}{6} + \frac{32}{3}

Simplify

= \frac{125}{6}

Graph

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Frequently Asked Questions (FAQ)

What is area x=y,x=-y^2+6y ?
The answer to area x=y,x=-y^2+6y is 125/6