Question 1

Which formula correctly gives the area $$A of a $$rectangle with length $$l$$ and width $$w$$?

Accepted Answer

$$A = lw$$

Question 2

A triangle has base $$b$$ and height $$h. $$What is the formula for its area $$A$$?

Accepted Answer

$$A = \frac{1}{2}bh$$

Question 3

How can the area of a polygon be found if it is not a standard shape?

Accepted Answer

By dividing it into triangles and summing their areas

Question 4

Given the region $$R$$ under $$y = 1 - x^2$ from x = 0$ to x = 1$, why can't its area be found by a simple formula?

Accepted Answer

Because the upper boundary is a curve, not a straight line

Question 5

When estimating the area under a curve using rectangles, what happens as the number of rectangles increases?

Accepted Answer

The estimate approaches the true area

Question 6

In the context of Riemann sums, what does the width $$\Delta x_k of $$the $$kth $$subinterval represent?

Accepted Answer

The difference $$x_k - x_{k-1}$$

Question 7

Which of the following best describes the lower sum in a Riemann sum approximation?

Accepted Answer

It uses the minimum value of the function on each subinterval

Question 8

Suppose you use right endpoints to estimate the area under $$y = x^2$ from x = 0$ to x = 1$ with n$ rectangles. What is the width of each rectangle?

Accepted Answer

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

Question 9

If the number of rectangles used in a Riemann sum increases to infinity, what does the sum approach?

Accepted Answer

The exact area under the curve

Question 10

Which of the following is the correct sigma notation for the sum of the areas of $$n$$ rectangles of width $$\Delta x$$ and heights $$f(x_k)$$?

Accepted Answer

$$\sum_{k=1}^n f(x_k) \Delta x$$

Question 11

What is the main advantage of using the midpoint rule for estimating the area under a curve?

Accepted Answer

It balances overestimates and underestimates, often giving a better approximation

Question 12

Given the function $$f(x) = x^2$ on $$[0,1]$$, what is the value of the right Riemann sum R_n$ as n$$ 	o \infty$$?

Accepted Answer

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

Question 13

If you want to estimate the area under $$y = \sin x$$ from $$x = 0 to$$ $$x = \pi$$ using rectangles, what happens to the upper sum estimate as the number of subintervals increases?

Accepted Answer

It approaches the exact average value

Question 14

Which of the following is true about the average value of a continuous function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

It is the area under $$f(x)$$ divided by $$b-a$$

Question 15

Why do finer partitions (more rectangles) give a better approximation of the area under a curve?

Accepted Answer

Because the rectangles better match the curve, reducing the error

Area and Estimating with Finite Sums: Foundations of Integration

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