Arc length and surface area of revolution Quiz

1. Which integral gives the arc length of $y=x^2$ from $x=0$ to $x=1$?

• A. $\int_0^1\sqrt{1+x^4}\,dx$
• B. $\int_0^1\sqrt{1+4x^2}\,dx$
• C. $\int_0^1(1+4x^2)\,dx$
• D. $\int_0^1\sqrt{1+2x}\,dx$

2. Find the length of the curve $x=3t$, $y=4t$ for $0\le t\le2$.

• A. $5$
• B. $14$
• C. $2$
• D. $10$

3. The line $y=2x$, $0\le x\le1$, is rotated about the $x$-axis. Find the surface area.

• A. $2\sqrt5\,\pi$
• B. $2\pi$
• C. $\frac{4\pi}{3}$
• D. $\sqrt5$

4. $y=x^2$, $0\le x\le1$, is rotated about the $y$-axis. Which integral gives the surface area?

• A. $2\pi\int_0^1x^2\sqrt{1+4x^2}\,dx$
• B. $2\pi\int_0^1x\sqrt{1+2x}\,dx$
• C. $2\pi\int_0^1x\sqrt{1+4x^2}\,dx$
• D. $\pi\int_0^1x^4\,dx$

5. $y=\sqrt{4-x^2}$, $-2\le x\le2$, is rotated about the $x$-axis to form a sphere. Find its surface area.

• A. $\frac{32\pi}{3}$
• B. $16\pi$
• C. $8\pi$
• D. $4\pi$