Parametric integration
Objective. Calculate areas using x(t), y(t) and dx/dt.
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Learn: the key idea
Parametric integration calculates the area under a curve defined by x(t) and y(t) using ∫y dx=∫y(dx/dt) dt, replacing the Cartesian differential dx with its parametric equivalent before integrating with respect to t. Example: for x=t², y=t, with t from 0 to 2, the area is ∫₀² t(2t) dt=∫₀² 2t² dt=[2t³/3]₀²=16/3. The limits must be converted from x values to the corresponding t values before integrating.
Worked example
Find the area under the parametric curve x=t², y=3t between t=0 and t=1 using parametric integration.
- 1Write the area integral as ∫y(dx/dt) dt with limits t=0 to t=1
- 2Differentiate x: dx/dt=2t, so integrand becomes 3t×2t=6t²
- 3Integrate: ∫₀¹ 6t² dt=[2t³]₀¹
- 4Evaluate the limits: 2(1)³−2(0)³=2
Answer: 2
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Start practisingCommon mistake. Students integrate y with respect to x directly using parametric formulas without replacing dx with (dx/dt)dt, producing an integral that cannot be evaluated correctly in terms of t.
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