Question

Use the fundamental theorem of calculus to find the area of the region bounded by the x-axis and the graph of y=4x^3-4x
Answer =

Answer 1

we need the x-intercepts.
4x^3 - 4x = 0
4x(x^2 - 1) 0
4x(x+1)(x-1) = 0
x = 0, -1, +1

If you sketch the curve, you will see two identical loops of the curve
cutting the x-axis, one above and one below.
The area of the loop at the left is
∫ (4x^3 - 4x) dx from -1 to 0
= [x^4 - 2x^2] from -1 to 0
= 0 - (1 - 2) = 1
so the area of the region bounded by the curve and the x-axis
= 2(1)
= 2 units^2