Arc length formula for a parametric curve?

s=\int{t1}^{t2}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt

Arc length formula for a polar curve?

s=\int{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta

Surface area for rotation about the x-axis, Cartesian?

S=2\pi\int y\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx

Surface area for rotation about the y-axis, parametric?

S=2\pi\int x\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt

What does ds represent?

A small element of arc length, \sqrt{dx^2+dy^2}.

Why is the surface element 2\pi y\,ds?

It is a thin band: circumference 2\pi y times width ds.

Key identity used for cardioid arc length?

1+\cos\theta=2\cos^2\frac{\theta}{2}

Arc length and area of a surface of revolutionEdexcel A-Level Further Maths: Flashcards

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Arc length formula for $y=f(x)$?

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Arc length formula for $y=f(x)$ ?: $s=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$
Arc length formula for a parametric curve?: $s=\int_{t_1}^{t_2}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$
Arc length formula for a polar curve?: $s=\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$
Surface area for rotation about the $x$ -axis, Cartesian?: $S=2\pi\int y\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$
Surface area for rotation about the $y$ -axis, parametric?: $S=2\pi\int x\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$
What does $ds$ represent?: A small element of arc length, $\sqrt{dx^2+dy^2}$ .
Why is the surface element $2\pi y\,ds$ ?: It is a thin band: circumference $2\pi y$ times width $ds$ .
Length of $y=\frac23x^{\frac32}$ from $x=0$ to $x=3$ ?: $\frac{14}{3}$
Distance from the initial line to a polar point $(r,\theta)$ ?: $y=r\sin\theta$
Key identity used for cardioid arc length?: $1+\cos\theta=2\cos^2\frac{\theta}{2}$
Length of the cardioid $r=2(1+\cos\theta)$ , $0\leq\theta\leq\pi$ ?: $8$
Surface area when $y=\sqrt{x}$ , $0\leq x\leq2$ , rotates about the $x$ -axis?: $\frac{13\pi}{3}$

Exam questions on Arc length and area of a surface of revolution

The curve $C$ has equation $y=\frac23x^{\frac32}$ for $x\geq0$ .
Find the exact length of the arc of $C$ from $x=0$ to $x=3$ .2 marks
A curve is given parametrically by $x=3t^2$ , $y=2t^3$ for $0\leq t\leq1$ .
Find the exact length of the curve.2 marks
The curve $C$ has equation $y=\sqrt{x}$ for $0\leq x\leq2$ . The arc of $C$ is rotated through $2\pi$ radians about the $x$ -axis to form a curved surface of area $S$ .
Show that $S=2\pi\int_0^2\sqrt{x+\frac14}\,dx$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).