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 $$ith $$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

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