Question

Find the area between y=x^3 and y=x at [-1,1]

Answer 1

0

Explanation:

`\int\limits^1_1 x^3 \, dx` -
`\int\limits^1_1 {x} \, dx`

Limits should be -1 to 1, its the integral of x^3 from -1 to 1 minus the integral of x from -1 to 1.

0-0 = 0

Answer 2

Over [-1, 0], you have x ³ ≥ x, and over [0, 1], x ³ ≤ x. So the area of the region is given by

∫₋₁⁰ (x ³ - x ) dx + ∫₀¹ (x - x ³) dx

= (1/4 x ⁴ - 1/2 x ²)|₋₁⁰ + (1/2 x ² - 1/4 x ⁴)|₀¹

= ((0 - 0) - (1/4 - 1/2)) + ((1/2 - 1/4) - (0 - 0))

= 1/2