$Find the area bounded by the graphs of two functions$

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$Find the area bounded by the graphs of two functions$

Find the area bounded by the graphs of the functions $y=x^2$ and $y=x^3$.

Calculus Integrals Geometry

Daniel90

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First find the intersection points of the two curves:
\[y=x^2, y=x^3 \Rightarrow x^2=x^3 \]
\[x^3-x^2=0 \Rightarrow x^2(x-1)=0 \Rightarrow x=0, 1.\]
Note that for $x$ between $0$ and $1$, we have $x^2 >x^3$. Thus the area bounded between the two graphs is
\[A=\int_0^1 x^2-x^3 dx=(\frac{x^3}{3}-\frac{x^4}{4})|_0^1=(\frac{1}{3}-\frac{1}{4})=\frac{1}{12}.\]

Rosetta

$Find the area bounded by the graphs of two functions$

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