Integration as the limit of a sum Quiz

1. Rectangles of width 1 with left-hand heights approximate the area under $y=x^2$ for $0\le x\le3$. The result is:

• A. an overestimate, $14$
• B. exactly $9$
• C. an underestimate, $5$
• D. an underestimate, $14$

2. Which integral is $\lim_{\delta x\to0}\sum_{x=2}^{5}x^2\,\delta x$?

• A. $\int_0^5x^2\,dx$
• B. $\int_2^5x^2\,dx$
• C. $\int_2^5x\,dx$
• D. $\int_2^6x^2\,dx$

3. For $n$ equal strips on $[0,6]$, the width of each strip is:

• A. $\frac6n$
• B. $6n$
• C. $\frac n6$
• D. $\frac{6}{n+1}$

4. A sum of rectangle areas is $4+\frac8n$ and tends to the exact area as $n\to\infty$. The exact area is:

• A. $12$
• B. $8$
• C. $\frac{8}{n}$
• D. $4$

5. A rod has density $\rho(x)=3$ kg per metre for $0\le x\le5$. Its mass is:

• A. $3$ kg
• B. $8$ kg
• C. $15$ kg
• D. $\frac35$ kg