Math integrals
posted by Melanie .
What is the area of the region bounded by y=x^2, the tangent to this parabola at (1, 1) and the xaxis?
Since it says that the parabola passes through 1,1 can I assume that the line is y = x or is that completely wrong?
Doing that I got ∫ x  x^2 dx (where the interval from a to b goes from 0 to 1)
so the solved integral would be
x^2/2  x^3/3] 0 to 1
= [(1)^2/2  (1)3/3]  0
= 1/2  1/3
= 1/6
What should I have done if it's wrong?

The tangent line is not y = x
Since the derivative of y = x^2 is dy/dx = 2x, at (1,1) dy/dx = 2, so the tangent line has a slope of 2
its equation would be y = 2x + b, with (1,1) on it
So, 1 = 2(1) + b, > b = 1
the tangent equation is y = 2x1, which would result in an xintercept of (1/2,0).
I would take the area between y = x^2 from 0 to 1 minus the right angled triangle formed by the xaxis, y=2x1 and x=1
let me know if you got 1/12.