Arc length and surface area of revolution Quiz

1. The length of the curve $y=f(x)$ from $x=a$ to $x=b$ is

• A. $\int_a^b\left(1+\left(\frac{dy}{dx}\right)^2\right)dx$
• B. $\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$
• C. $\int_a^b\sqrt{1+\frac{dy}{dx}}\,dx$
• D. $\int_a^b\sqrt{\left(\frac{dy}{dx}\right)^2}\,dx$

2. For $x=t^2$, $y=\frac{t^3}{3}$ with $t\ge0$, the arc length integrand is

• A. $\sqrt{2t+t^2}$
• B. $\sqrt{4+t^2}$
• C. $t\sqrt{4+t^2}$
• D. $t^2\sqrt{4+t^2}$

3. The line $y=2x$ for $0\le x\le3$ is rotated about the $x$-axis. The surface area is

• A. $9\sqrt5\,\pi$
• B. $18\pi$
• C. $6\sqrt5\,\pi$
• D. $18\sqrt5\,\pi$

4. Rotating about the $y$-axis, the surface area is $2\pi\int(\ldots)\,ds$ where $(\ldots)$ is

• A. $y$
• B. $x$
• C. $\pi x^2$
• D. $\pi y^2$

5. In $S=\int2\pi y\,ds$, the factor $2\pi y$ is

• A. the circumference traced by a point at height $y$
• B. the area of a thin strip
• C. the volume of a slice
• D. the length of the arc