Improper integrals and the mean value of a function Quiz

1. Find $\int_1^\infty x^{-3}\,dx$.

• A. $\frac13$
• B. $\frac12$
• C. $1$
• D. It diverges

2. Which of these integrals converges?

• A. $\int_1^\infty\frac1x\,dx$
• B. $\int_0^1\frac1x\,dx$
• C. $\int_0^1x^{-1/2}\,dx$
• D. $\int_1^\infty x^{-1/2}\,dx$

3. Find the mean value of $f(x)=x^2$ over $1\le x\le4$.

• A. $21$
• B. $\frac{15}{2}$
• C. $\frac{63}{4}$
• D. $7$

4. Find $\int_0^\infty e^{-3x}\,dx$.

• A. $\frac13$
• B. $3$
• C. $-\frac13$
• D. It diverges

5. What is the value of $\int_{-1}^1\frac{1}{x^2}\,dx$?

• A. $-2$
• B. $2$
• C. $0$
• D. It diverges