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Integration of Parametric, Polar, and Vector Curves for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

Area for Parametric Curves

For a curve defined parametrically by x = f(t) and y = g(t), the area bounded by the portion of the curve between t = α and t = β is

Example 1

Find the area bounded by x = 2sin t, y = 3sin2 t.

Step 1: Determine the limits of integration. The symmetry of the graph allows us to integrate from t = 0 to t = π/2 and multiply by 2.

Step 2: Differentiate

Step 3:

Arc Length for Parametric Curves

The length of that arc is

Example 2

Find the length of the arc defined by x = et cos t and y = et sin t from t = 0 to t = 4.

Step 1: Differentiate

Step 2:

Surface Area for Parametric Curves

The surface area created when that arc is revolved about the x-axis is

Example 3

Find the area of the surface generated by revolving about the x-axis the arc defined by x =3 – 2t and when 0 ≤ t ≤ 4.

Step 1: Differentiate

Step 2:

Area for Polar Curves

If r = f (θ) is a continuous polar curve on the interval α ≤ θ ≤ β and α<β<α + 2π, then the area enclosed by the polar curve is

Example 1

Find the area enclosed by r = 2 + 2 cos θ on the interval from θ = 0 to θ = π.

Step 1: Square r2 = 4 + 8 cos θ + 4 cos2 θ.

Step 2:

Arc Length for Polar Curves

For a polar graph defined on a interval (α, β), if the graph does not retrace itself in that interval and if is continuous, then the length of the arc from θ =α to θ =β is

Example 2

Find the length of the spiral r = eθ from θ = 0 to θ = π.

Step 1: Differentiate

Step 2: Square r2 =e.

Step 3:

Integrating a Vector Function

For a vector-valued function

Example 1

The acceleration vector of a particle at any time If at time t = 0, its velocity is i + j and its displacement is 0, find the functions for the position and velocity at any time t.

Length of a Vector Curve

The length of a curve defined by the vector-valued function

Example 2

Find the length of the curve

Step 3: With the aid of a graphing calculator, the arc length can be found to be approximately equal to 16.319 units.

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

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