The idea

Areas under parametric curves

Finding the area under a curve given parametrically as ∫ y (dx/dt) dt with the limits in t — and combining it with tangents/normals to find regions bounded by the curve, a line and an axis.

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Pure · Integration

Areas under parametric curves

Finding the area under a curve given parametrically as ∫ y (dx/dt) dt with the limits in t — and combining it with tangents/normals to find regions bounded by the curve, a line and an axis.

Why it works

Substituting t into the area integral

The area under a curve is

\displaystyle\int y\,dx

. When the curve is given parametrically —

x = f(t)

y = g(t)

— both

y

and

dx

need to be put in terms of

t

before you can integrate:

\int y\,dx = \int_{t_1}^{t_2} y\,\frac{dx}{dt}\,dt

A first computation

Take the curve

x = t^2,\ y = 4t

(for

t \ge 0

). To find the area between it and the

x

-axis from

x = 0

x = 4

, note these are

t = 0

and

t = 2

. With

\dfrac{dx}{dt} = 2t

\int y\,\frac{dx}{dt}\,dt = \int_0^2 4t\cdot 2t\,dt = \int_0^2 8t^2\,dt = \left[\tfrac83 t^3\right]_0^2 = \frac{64}{3}.

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