Take this Quiz: Calculus AB — integration and the Fundamental Theorem | Zestly

Question 1

A function $f$ is strictly increasing on $[0, 4]$, with $f(0) = 1$, $f(2) = 3$ and $f(4) = 7$. Using a right Riemann sum with the two subintervals $[0, 2]$ and $[2, 4]$, what is the approximation of $\int_0^4 f(x)\,dx$, and how does it compare with the true value?

Accepted Answer

$20$, an over-estimate

Answer

$8$, an over-estimate

Answer

$20$, an under-estimate

Answer

$8$, an under-estimate

Question 2

Water flows into a tank at a rate of $r(t) = 3t^2 + 2$ litres per minute, where $t$ is measured in minutes. What does $\int_0^5 r(t) \, dt$ represent?

Accepted Answer

The volume of water added to the tank during the first five minutes, in litres

Answer

The volume of water standing in the tank at $t = 5$, in litres

Answer

The rate at which the volume is changing at $t = 5$, in litres per minute

Answer

The average rate of inflow over the five minutes, in litres per minute

Question 3

Let $F(x) = \int_1^{x^2} \sin(t^2) \, dt$. What is $F'(x)$?

Accepted Answer

$2x \sin(x^4)$

Answer

$\sin(x^2)$

Answer

$\sin(x^4)$

Answer

$x^2 \sin(x^2)$

Question 4

Evaluate $\int_0^2 (3x^2 - 1) \, dx$.

Answer

$8$

Answer

$10$

Answer

$12$

Answer

$6$

Question 5

Consider $f(x) = x^3$ on the interval $[-1, 1]$. What is the value of $\int_{-1}^{1} x^3 \, dx$, and what is the total area enclosed between the graph and the $x$-axis on that interval?

Answer

The integral is $0$; the total area is $1/4$

Answer

The integral is $1/2$; the total area is $1/2$

Answer

The integral is $0$; the total area is $1/2$

Answer

The integral is $0$; the total area is also $0$