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The approximate area of the region bounded by the curves f(x) = x / √(x² + 1) and g(x) = x⁴ - x is approximately 0.806.
<h3>How to determine the approximate area of the regions bounded by the curves</h3>
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
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Answer:
Step-by-step explanation:
300 degrees is in the fourth quadrant (it's between 270 and 360); sine is negative in the fourth quadrant.
Given we're in the fourth quadrant, the reference angle is 360 - 300 = 60 degrees
sin(60°) =
And since sine is negative, this value turns negative:
sin(300°) =