Parametric equations, polar coordinates, and vector-valued functions

Arc length: parametric curves

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Parametric curve arc length
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Worked example: Parametric arc length
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Parametric curve arc lengthGet 3 of 4 questions to level up!
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Planar motion

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Planar motion (with integrals)
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Planar motion (with integrals)Get 3 of 4 questions to level up!
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Area: polar regions (single curve)

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Area bounded by polar curves
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Worked example: Area enclosed by cardioid
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Area bounded by polar curves introGet 3 of 4 questions to level up!
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Area bounded by polar curvesGet 3 of 4 questions to level up!
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Area: polar regions (two curves)

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Worked example: Area between two polar graphs
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Area between two polar curvesGet 3 of 4 questions to level up!
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Arc length: polar curves

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Arc length of polar curves
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Worked example: Arc length of polar curves
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Arc length of polar curvesGet 3 of 4 questions to level up!
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Calculator-active practice

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Evaluating definite integral with calculator
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Area with polar functions (calculator-active)Get 3 of 4 questions to level up!
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About this unit

We are used to working with functions whose output is a single variable, and whose graph is defined with Cartesian, i.e., (x,y) coordinates. But there can be other functions! For example, vector-valued functions can have two variables or more as outputs! Polar functions are graphed using polar coordinates, i.e., they take an angle as an input and output a radius! Learn about these functions and how we apply the concept of the integral on them.