Q: Dr. Austin checks her speedometer every hour for 3 hours, recording speeds of 0, 18, 54, and 63 mph. Using the left endpoint approximation, what is the estimated total distance traveled?

$0 \times 1 + 18 \times 1 + 54 \times 1 = 72$ miles

Q: Which of the following is the correct definition of the definite integral?

$\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$

Question 1

Which of the following best describes the relationship between the area under a velocity-time curve and the distance traveled?

Accepted Answer

The area under the curve represents the total distance traveled.

Question 2

Suppose $v(t)$ is a velocity function. Which formula correctly expresses the distance traveled from $t=a$ to $t=b$?

Accepted Answer

$\int_a^b v(t) \, dt$

Question 3

When approximating the area under a curve using rectangles, what is the name of the method where the height of each rectangle is determined by the left endpoint of each subinterval?

Accepted Answer

Left Riemann Sum

Question 4

Given $f(x) = x^2$ on $[0,2]$, which expression represents the area approximation using 2 rectangles and right endpoints?

Accepted Answer

$f(1) \Delta x + f(2) \Delta x$

Question 5

If the interval $[a,b]$ is divided into $n$ subintervals of equal width, what is the formula for the width $\Delta x$ of each subinterval?

Accepted Answer

$\Delta x = \frac{b-a}{n}$

Question 6

Which of the following statements about the definite integral $\int_a^b f(x) \, dx$ is TRUE?

Accepted Answer

It represents the limit of Riemann sums as the number of subintervals approaches infinity.

Question 7

A biologist wants to estimate the area under the curve $f(x) = 2x$ from $x=0$ to $x=4$ using 4 rectangles and the midpoint rule. What is the width of each rectangle?

Accepted Answer

$\Delta x = 1$

Question 8

Which of the following is a key advantage of using more rectangles in a Riemann sum approximation?

Accepted Answer

It generally increases the accuracy of the area approximation.

Question 9

Dr. Austin checks her speedometer every hour for 3 hours, recording speeds of 0, 18, 54, and 63 mph. Using the left endpoint approximation, what is the estimated total distance traveled?

Accepted Answer

$0 	imes 1 + 18 	imes 1 + 54 	imes 1 = 72$ miles

Question 10

Which of the following best describes the midpoint rule for approximating the area under a curve?

Accepted Answer

The height of each rectangle is the value of the function at the midpoint of each subinterval.

Question 11

If $f(x)$ is negative over $[a,b]$, what can be said about $\int_a^b f(x) \, dx$?

Accepted Answer

It will be negative.

Question 12

Which of the following is the correct definition of the definite integral?

Accepted Answer

$\int_a^b f(x) \, dx = \lim_{n 	o \infty} \sum_{i=1}^n f(x_i^*) \Delta x$

Question 13

A student uses the right endpoint rule to approximate $\int_0^2 x^2 \, dx$ with 2 rectangles. What are the $x$-values used to determine the heights?

Accepted Answer

$x=1$ and $x=2$

Question 14

Which of the following is NOT a valid way to choose the height of rectangles in a Riemann sum?

Accepted Answer

Using the average of all function values on $[a,b]$

Question 15

If the actual area under $f(x) = 2x$ from $x=0$ to $x=4$ is calculated using geometry, what is the value?

Accepted Answer

$8$

Question 16

Which of the following best explains why the midpoint rule can be more accurate than the left or right endpoint rules?

Accepted Answer

It tends to balance overestimates and underestimates, reducing error.

Question 17

Suppose you approximate $\int_0^2 x^2 \, dx$ using 2 rectangles and the left endpoint rule. What is the approximation?

Accepted Answer

$0 	imes 1 + 1 	imes 1 = 1$

Question 18

Which of the following is the correct expression for the area of all rectangles in a Riemann sum?

Accepted Answer

$\sum_{i=1}^n f(x_i^*) \Delta x$

Question 19

If the subintervals are not of equal width, which formula should be used to approximate the area under the curve?

Accepted Answer

$\sum_{i=1}^n f(x_i^*) \Delta x_i$

Area and Definite Integrals: Riemann Sums and Applications

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