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Area, Volume, and Arc Length

Area, Volume, and Arc Length

Example 1

Write an integral expression for the length of the curve described by the parametric equations x(t) = et and y(t) = 2t + 1 for 0 ≤ t ≤ 4.

Example 2

The equations x(t) = cos t and y(t) = 2sin t describe a parametric curve.

(a) Write an integral expression for the length of the portion of the curve graphed below.

(b) Use a calculator to evaluate your integral and explain why your answer is reasonable.

Example 3

Write an integral expression for the parametric function described by the equations

x(t) = x3

y(t) = etsin t

for 0 ≤ t ≤ π.

Example 4

Write an integral expression for the length of the parametric curve

x = t cos t

y = t sin t

for α ≤ t ≤ β. Use the equality sin2 t + cos2 t = 1 to simplify your expression. (It will still be an integral.)

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