Question 1

What is the purpose of partitioning the interval $$[0,1]$$ when finding the area under the curve $$f(x) = x^2$?

Accepted Answer

To approximate the area using rectangles whose total area approaches the true area as the partition gets finer.

Question 2

Given the partition $$\{0, \frac{1}{2}, 1\} of$$ $$[0,1]$$, what is the width $$\Delta x of $$each subinterval?

Accepted Answer

$$\Delta x = \frac{1}{2}$$

Question 3

Which of the following best describes the lower sum $$L_n(f)$$ for $$f(x) = x^2$ on $$[0,1]$$?

Accepted Answer

$$L_n(f) is $$the sum of the areas of rectangles using the left endpoint of each subinterval.

Question 4

What is the formula for the sum of the squares of the first $$n$$ natural numbers?

Accepted Answer

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

Question 5

If $$L_2(f$$) \leq A \leq $$U_2(f$$)$$, what does A$ represent?

Accepted Answer

$$A is $$the actual area under $$f(x) on$$ $$[0,1].$$

Question 6

As the number of subintervals $$n$$ increases, what happens to the difference between the upper sum $$U_n(f)$$ and the lower sum $$L_n(f)$$ for $$f(x) = x^2$ on $$[0,1]$$?

Accepted Answer

The difference approaches zero.

Question 7

Given $$L_n(f) = \frac{1}{3} - \frac{1}{2n} + \frac{1}{6n^2$}$$, what is $$\lim_{n 	o \infty} L_n(f)$$?

Accepted Answer

$$\frac{1}{3}$$

Question 8

Which of the following expressions gives the upper sum $$U_n(f)$$ for $$f(x) = x^2$ on $$[0,1]$$?

Accepted Answer

$$U_n(f) = \frac{1}{3} + \frac{1}{2n} + \frac{1}{6n^2$}$$

Question 9

Why do we use both lower and upper sums when approximating the area under a curve?

Accepted Answer

To bound the true area from below and above, improving accuracy as partitions get finer.

Question 10

Suppose you partition $$[0,1]$$ into $$n$$ equal subintervals. What is the width of each subinterval?

Accepted Answer

$$\frac{1}{n}$$

Question 11

If $$f(x) = x^2$ and you use the right endpoint for each subinterval in $$[0,1]$$ with n$ subintervals, which sum are you calculating?

Accepted Answer

The upper sum $$U_n(f)$$

Question 12

What is the value of $$\$$int_0$$^1 x^2$ dx$$ based on the limit of the lower and upper sums?

Accepted Answer

$$\frac{1}{3}$$

Question 13

Which of the following best describes a partition of an interval?

Accepted Answer

A set of points that divides the interval into subintervals.

Question 14

If you increase the number of rectangles used to approximate the area under $$y = x^2$ on $$[0,1]$$, what happens to the approximation?

Accepted Answer

It becomes more accurate.

Question 15

Which real-world scenario can be modeled by finding the area under a curve using definite integrals?

Accepted Answer

Calculating the total distance traveled by a car given its velocity over time.

Definite Integrals: Area Under a Curve and Riemann Sums

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