Just an integration question

Find the equation of the tangent at the point (1,0) to the curve y=(x-1)(x^2-1). Find the area of the finite region enclosed by the curve and the tangent.

Question

Just an integration question

Find the equation of the tangent at the point (1,0) to the curve y=(x-1)(x^2-1). Find the area of the finite region enclosed by the curve and the tangent.

anonymous · Answer

$$y'=(x^2-1)+(x-1)(2x)=3x^2-2x-1$$
The slope at $(1,0)$ will be $y'(1)=3-2-1=0$.

Find the equation of the line with slope 0 and that passes through (1,0).

Then you'll need the intersection of tangent with the curve (there should be two points of intersection) which will help you determine the limits of integration. The integrand depends on which is greater - the line or the curve - over the limits.

anonymous · Answer

A drawing might help clear this up. Since $y=(x-1)(x+1)(x-1)=(x-1)^2(x+1)$, you have
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