Centres of mass of uniform bodies by integration Quiz

1. Find $\bar x$ for the region under $y=x^2$ from $x=0$ to $x=3$.

• A. $\frac32$
• B. $\frac{27}{4}$
• C. $3$
• D. $\frac94$

2. Find the distance of the centre of mass of a solid hemisphere of radius $6$ cm from its plane face.

• A. $2.25$ cm
• B. $3$ cm
• C. $2$ cm
• D. $4$ cm

3. A solid cone has height $20$ cm. How far is its centre of mass from the vertex?

• A. $5$ cm
• B. $10$ cm
• C. $15$ cm
• D. $\frac{40}{3}$ cm

4. A uniform square lamina $0\le x\le6$, $0\le y\le6$ (cm) has the square $0\le x\le2$, $0\le y\le2$ removed. Find $\bar x$ for the remaining lamina.

• A. $3.5$ cm
• B. $3.25$ cm
• C. $3$ cm
• D. $2.9$ cm

5. A solid is formed by rotating $y=f(x)$ about the $x$-axis. Which expression gives $\bar x$?

• A. $\frac{\int xy\,dx}{\int y\,dx}$
• B. $\frac{\int xy^2\,dx}{\int y^2\,dx}$
• C. $\frac{\frac12\int y^2\,dx}{\int y\,dx}$
• D. $\frac{\int y^2\,dx}{\int xy^2\,dx}$