Question

Find the approximate area of the regions bounded by the curves y = x/(√x2+ 1) and y = x^4−x. (You may use the points of intersection x = 0, x ≈1.18 of the two curves.)

Answer 1

The approximate area of the region bounded by the curves f(x) = x / √(x² + 1) and g(x) = x⁴ - x is approximately 0.806.

How to determine the approximate area of the regions bounded by the curves

In this problem we must use definite integrals to determine the area of the region bounded by the curves. Based on all the information given by the graph attached below, the area can be defined in accordance with this formula:

A = A₁ + A₂                                                                (1)

A₁ = ∫ [g(x) - f(x)] dx, for x ∈ [- 0.786, 0]                   (2)

A₂ = ∫ [f(x) - g(x)] dx, for x ∈ [0, 1.151]                       (3)

g(x) = x⁴ - x                                                               (4)

f(x) = x / √(x² + 1)                                                      (5)

Then, we proceed to find the integrals:

∫ g(x) dx = ∫ x⁴ dx - ∫ x dx = (1 / 5) · x⁵ - (1 / 2) · x²                          (6)

∫ f(x) dx = ∫ [x / √(x² + 1)] dx = (1 / 2) ∫ [2 · x / √(x² + 1)] dx = (1 / 2) ∫ [du / √u] = √u = √(x² + 1)                                                                                  (7)

And the complete expression for the integral is:

A = A₁ + A₂                                                                                      (1b)

A₁ = (1 / 5) · x⁵ - (1 / 2) · x² - √(x² + 1), for x ∈ [- 0.786, 0]               (2b)

A₂ = √(x² + 1) - (1 / 5) · x⁵ + (1 / 2) · x², for x ∈ [0, 1.151]                  (3b)

A₁ = 0.023

A₂ = 0.783

A = 0.023 + 0.783

A = 0.806

The approximate area of the region bounded by the curves f(x) = x / √(x² + 1) and g(x) = x⁴ - x is approximately 0.806.

To learn more on definite integral: brainly.com/question/14279102

#SPJ1