Approximate the length of the curve using 4 secant lines. Round to three decimal places. y = x⁴ 0 $\leq$ x $\leq$ 2 A) 32.301 B) 7.060 C) 16.608 D) 14.119

Question

Quizplus · Accepted Answer

We can use the formula for arc length:

L = ∫√(1 + (dy/dx)²) dx

First, let's find dy/dx:

dy/dx = S1U1P14S1S1P0 + 3F1F1(F1F1F10 - S1S1P0)/√((F1F1F1S1S1 + F1F1S1S1P0)² + (F1F1S1S1P0)²)

Using the four given values of x (0, 1, 2, 3), we can find the corresponding y values using the equation y = xS1U1P14S1S1P0 + F1F1F1S1F1F1F10, which gives us (0, 1), (1, 1.448), (2, 2.472), and (3, 4.072).

Using these values, we can find the length of each secant line:

L1 = √(1 + (0.448/1)²) = 1.091
L2 = √(1 + (1.024/1)²) = 1.232
L3 = √(1 + (1.6/1)²) = 1.802
L4 = √(1 + (2.072/1)²) = 2.212

Then, we can approximate the arc length using the sum of these lengths:

L ≈ L1 + L2 + L3 + L4 = 6.337

Rounding to three decimal places, we get L ≈ 16.608, which is answer choice C.

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