Q: Which of the following is the correct formula for the mesh of a partition $$P of$$ $$[a, b]$$?

$$\text{mesh}(P) = \max \{ \Delta x_i \}$$

Q: If $$f(x) is $$continuous and non-negative on $$[a, b]$$, which of the following is always true about the area under $$f(x)$$ from $$a to$$ $$b$$?

It is equal to $$\lim_{n \to \infty} R_n$$ for any choice of sample points in each subinterval.

Question 1

Given the region $$R$$ bounded by $$y = x^2$, y = 0$, x = 1$, and x = 3$, which of the following expressions correctly represents the right-endpoint Riemann sum R_n$ for approximating the area of R$ using n$ subintervals of equal width?

Accepted Answer

$$R_n = \sum_{i=1}^n \left(1 + \frac{2i}{n}\$$right)^2$$ \cdot \frac{2}{n}$$

Question 2

What is the width $$\Delta x of $$each subinterval when dividing the interval $$[1,3]$$ into $$n$$ equal parts for a Riemann sum?

Accepted Answer

$$\Delta x = \frac{2}{n}$$

Question 3

Suppose you use a left-endpoint Riemann sum $$L_n to $$approximate the area under $$y = x^2$ from x = 1$ to x = 3$. Which of the following expressions gives L_n$?

Accepted Answer

$$L_n = \sum_{i=1}^n \left(1 + \frac{2(i-1)}{n}\$$right)^2$$ \cdot \frac{2}{n}$$

Question 4

Evaluate $$\lim_{n 	o \infty} R_n$$ for the right-endpoint Riemann sum for the area under $$y = x^2$ from x = 1$ to x = 3$.

Accepted Answer

$$\frac{26}{3}$$

Question 5

Which of the following best describes a partition $$P of $$the interval $$[a, b]$$?

Accepted Answer

A set of points $$a = x_0$ \u003c x_1$ \u003c x_2$ \u003c \ldots \u003c x_n = b$$ dividing $$[a, b]$$ into subintervals

Question 6

If a partition of $$[0,4] is $$given by the points $$0, 1, 3, 5, 4$$, which of the following is true?

Accepted Answer

This is not a valid partition because the points are not in increasing order.

Question 7

Given a function $$f(x) \geq 0 on$$ $$[a, b]$$, which of the following Riemann sums will always underestimate the area under $$f(x) if$$ $$f(x) is $$increasing on $$[a, b]$$?

Accepted Answer

Left-endpoint Riemann sum

Question 8

For the function $$f(x) = x^2$ on $$[1,3]$$, what is the value of the definite integral $$\$$int_1^3$$ $$x^2$$ \, dx$$?

Accepted Answer

$$\frac{26}{3}$$

Question 9

Which of the following statements about Riemann sums is TRUE?

Accepted Answer

As the mesh of the partition approaches zero, the Riemann sum approaches the definite integral.

Question 10

Suppose you are approximating the area under $$y = x^2$ from x = 1$ to x = 3$ using a right-endpoint Riemann sum with n$ subintervals. What is the i$th right endpoint x_i$?

Accepted Answer

$$x_i = 1 + \frac{2i}{n}$$

Question 11

Which of the following is the correct formula for the mesh of a partition $$P of$$ $$[a, b]$$?

Accepted Answer

$$	ext{mesh}(P) = \max \{ \Delta x_i \}$$

Question 12

A Riemann sum for $$f(x)$$ associated to a partition $$P is $$given by which of the following?

Accepted Answer

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

Question 13

If $$f(x) is $$continuous and non-negative on $$[a, b]$$, which of the following is always true about the area under $$f(x)$$ from $$a to$$ $$b$$?

Accepted Answer

It is equal to $$\lim_{n 	o \infty} R_n$$ for any choice of sample points in each subinterval.

Question 14

Which of the following is NOT a requirement for a valid partition of $$[a, b]$$?

Accepted Answer

All subintervals must have equal width.

Question 15

In a real-world scenario, suppose the rate of water flow into a tank is given by $$f(t)$$ liters per hour, where $$t is in $$hours. Which Riemann sum would best approximate the total water added between $$t = 2$$ and $$t = 6 if $$measurements are taken at the end of each hour?

Accepted Answer

Right-endpoint Riemann sum

Question 16

If $$f(x) is $$decreasing on $$[a, b]$$, which Riemann sum will overestimate the area under $$f(x)$$?

Accepted Answer

Left-endpoint Riemann sum

Question 17

Which of the following is the correct formula for the sum of squares $$\sum_{i=1}^n i^2$?

Accepted Answer

$$\sum_{i=1}^n i^2$ = \frac{n(n+1)(2n+1)}{6}$$

Question 18

A student claims that for any function $$f(x)$$, the left-endpoint and right-endpoint Riemann sums will always be equal. Is this statement correct?

Accepted Answer

No, unless $$f(x) is $$constant on $$[a, b].$$

Question 19

Suppose you are using a Riemann sum to estimate the area under a curve representing the velocity of a car over time. What does the Riemann sum represent in this context?

Accepted Answer

The total distance traveled by the car over the time interval.

Riemann Sums and Area Approximation: Foundations of Integration

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