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Calculate the area bounded by the curves

$y=x^2-1$ and $y=x-1$

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enter image description here

First note that the curves intersect at $(0,-1)$ and $(1,0)$. And the area you are interested in is the area between the two curves between these two points. Hence, we get the area between the two curves as $$\int_{x=0}^{x=1} \int_{y=x^2-1}^{y=x-1} dydx = \int_{x=0}^{x=1} (x-x^2) dydx = \left(\dfrac{x}2 - \dfrac{x^3}3 \right)_{x=0}^{x=1} = \dfrac12 - \dfrac13 = \dfrac16$$

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+1 nothing, but a graph, can illustrate a maths problem. – Babak S. Jan 20 '13 at 19:01

First, calculate the intersection of the functions.

$$ x - 1 = x^2 - 1 $$ $$ 0 = x^2 - x $$

We get: $$ x = 0, 1 $$

Now we integrate:

$$ \int_{0}^1 (x^2 - 1) - (x-1) \mathrm d x $$ $$ \int_{0}^1 (x^2 - x) \mathrm d x $$ $$ -\frac{1}{6} $$

Clearly the area must be positive, so we have: $$ \color{green}{\frac{1}{6}} $$

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Hint: Find the point of intersection of the two curves, calculate the area of each curve by integrating through the point of intersection, then subtract areas to get whats between them.

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