Question 1

Which of the following best describes the process of integration as introduced in the study materials?

Accepted Answer

Integration is the process of adding up a bunch of rectangles to estimate the area under a curve.

Question 2

Given the function $f(x) = 1$ on the interval $[0,1]$, what is the area under the curve?

Accepted Answer

$1$

Question 3

For $f(x) = x$ on $[0,1]$, what geometric shape does the area under the curve represent, and what is its area?

Accepted Answer

A triangle, area $\frac{1}{2}$

Question 4

Why can't we use a simple geometric formula to find the area under $f(x) = x^2$ from $0$ to $1$?

Accepted Answer

Because the curve is a parabola and there is no simple geometric formula for its area.

Question 5

What is the formula for the width of each rectangle in a Riemann sum approximation over $[a, b]$ using $n$ rectangles?

Accepted Answer

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

Question 6

If you use left endpoints to approximate the area under $f(x) = x^2$ from $0$ to $1$ with $3$ rectangles, what $x$-values do you use for the heights?

Accepted Answer

$0, \frac{1}{3}, \frac{2}{3}$

Question 7

Which of the following statements about Riemann sums is TRUE?

Accepted Answer

The more rectangles you use, the better your approximation of the area under the curve.

Question 8

Suppose you are approximating the area under $f(x) = x^2$ from $0$ to $1$ using $3$ rectangles and the right endpoint method. What is the approximate area?

Accepted Answer

$\frac{14}{27}$

Question 9

What is the general formula for a Riemann sum using $n$ rectangles for a function $f(x)$ on $[a, b]$?

Accepted Answer

$\sum\limits_{i=1}^{n} f(x_i) \Delta x$

Question 10

If you use the midpoint rule to approximate the area under $f(x) = x^2$ from $0$ to $1$ with $3$ rectangles, which $x$-values do you use for the heights?

Accepted Answer

$\frac{1}{6}, \frac{1}{2}, \frac{5}{6}$

Question 11

A biologist wants to estimate the area under the curve $y = \sin(x)$ from $x = 0$ to $x = \pi$ using $4$ rectangles and the left endpoint method. What is the width of each rectangle?

Accepted Answer

$\frac{\pi}{4}$

Question 12

What happens to the Riemann sum approximation as the number of rectangles $n$ approaches infinity?

Accepted Answer

The approximation approaches the exact area under the curve.

Question 13

Which formula would you use to find the $i$th left endpoint for a Riemann sum on $[a, b]$ with $n$ rectangles?

Accepted Answer

$x_i^* = a + (i-1) \Delta x$

Question 14

A social scientist uses a Riemann sum to estimate the area under $f(x) = x^2 - x$ from $0$ to $1$ with $6$ rectangles and left endpoints. What is $\Delta x$?

Accepted Answer

$\frac{1}{6}$

Question 15

Which of the following is a correct interpretation of the definite integral $\int_a^b f(x) dx$ based on the Riemann sum approach?

Accepted Answer

It is the limit of the sum of the areas of rectangles as the number of rectangles approaches infinity.

Question 16

If a Riemann sum uses right endpoints, which $x$-values are used for the heights of the rectangles?

Accepted Answer

The rightmost points of each subinterval.

Question 17

In a real-world context, why might a scientist use a Riemann sum instead of an exact integral?

Accepted Answer

Because the function may not have a simple antiderivative or the data is only available at discrete points.

Introduction to Integration and Riemann Sums

Study Guide - Practice Questions

Study Guide - Flashcards