Area between curves and integration as the limit of a sum Quiz

1. Find the area between $y=2x$ and $y=x^2$.

• A. $\frac43$
• B. $\frac83$
• C. $4$
• D. $\frac23$

2. A curve has $x=2t$, $y=t^2$. Find the area under it between $x=0$ and $x=4$.

• A. $\frac{8}{3}$
• B. $\frac{16}{3}$
• C. $\frac{64}{3}$
• D. $8$

3. As $\delta x\to0$, $\sum_{x=a}^{x=b}f(x)\,\delta x$ tends to which expression?

• A. $f(b)-f(a)$
• B. $f'(x)$
• C. $\int_a^bf(x)\,\mathrm{d}x$
• D. $\frac{f(b)}{f(a)}$

4. A function is decreasing. Rectangles use the height at the left-hand end of each strip. What do they give?

• A. An underestimate
• B. The exact area
• C. A value of zero
• D. An overestimate

5. A curve has $x=t^3$ and $y=t+1$. Which $t$-limits correspond to $x=1$ and $x=8$?

• A. $1$ and $8$
• B. $1$ and $2$
• C. $1$ and $512$
• D. $0$ and $2$