Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Calculate the area bounded by the curves

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

share|cite|improve this question

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 $$\int_{x=0}^{x=1} \int_{y=x^2-1}^{y=x-1} dydx = \int_{x=0}^{x=1} (x-x^2) dydx = \left(\dfrac{x}2 - \dfrac{x^3}3 \right)_{x=0}^{x=1} = \dfrac12 - \dfrac13 = \dfrac16$$

share|cite|improve this answer
+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: $$ \color{green}{\frac{1}{6}} $$

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.