Q: If $$f(x) is $$positive and decreasing on $$[a, b]$$, what happens to the error bound for left and right sums as the number of rectangles $$n$$ increases?

The error bound approaches zero as $$n \to \infty.$$

Question 1

Which of the following best describes the geometric interpretation of the definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

It represents the cumulative sum of the signed areas between the graph of $$f(x)$$ and the x-axis from $$x=a to$$ $$x=b$$, where areas above the x-axis are positive and areas below are negative.

Question 2

Given the function $$f(x) = 0.25x^2$ + 1 on $$the interval $$[1, 5]$$, which statement about the left sum $$L_4$$ and right sum $$R_4 is $$correct?

Accepted Answer

$$L_4 is an $$underestimate and $$R_4 is an $$overestimate of the actual area.

Question 3

If $$f(x) is $$positive and decreasing on $$[a, b]$$, what happens to the error bound for left and right sums as the number of rectangles $$n$$ increases?

Accepted Answer

The error bound approaches zero as $$n 	o \infty.$$

Question 4

For the function $$f(x) = 8 - 0.2x^2$ on $$[2, 5]$$, using n=6$ rectangles, what is the width $$\Delta x$$ of each rectangle?

Accepted Answer

$$\Delta x = 0.5$$

Question 5

Which of the following is a general Riemann sum for a function $$f on$$ $$[a, b]$$ partitioned into $$n$$ subintervals?

Accepted Answer

$$S_n = f(c_1$)\Delta x + f(c_2$)\Delta x + \cdots + f(c_n)\Delta x$$ where $$c_k is $$any point in $$[x_{k-1}, x_k]$$

Question 6

Suppose $$f(x) is $$continuous on $$[a, b]. $$What does Theorem 3 guarantee about the Riemann sums for $$f as$$ $$n 	o \infty$$?

Accepted Answer

The Riemann sums approach a real number limit $$I.$$

Question 7

In the context of definite integrals, what is the integrand in $$\int_a^b f(x)\,dx$$?

Accepted Answer

The function $$f(x)$$

Question 8

If a function $$f(x) is $$negative on a subinterval $$[x_{k-1}, x_k]$$, what does the corresponding term $$f(c_k)\Delta x in $$the Riemann sum represent?

Accepted Answer

A negative area below the x-axis.

Question 9

Given the areas in the figure: Area A = 2.12 (below x-axis), Area B = 11.9 (above x-axis), Area C = 7.67 (below x-axis), what is the value of $$\int_a^d f(x)\,dx$$?

Accepted Answer

$$-2.12 + 11.9 - 7.67$$

Question 10

Which of the following statements about left and right sums is true for a function $$f(x)$$ that is increasing on $$[a, b]$$?

Accepted Answer

The left sum underestimates the area and the right sum overestimates the area.

Question 11

If you want the approximation of the area under $$f(x) by$$ $$L_n$$ and $$R_n to be $$within $$0.05 of $$the true value, what should you do?

Accepted Answer

Increase the number of rectangles $$n$$ until the error bound is less than or equal to $$0.05.$$

Question 12

Which property of definite integrals allows you to split the integral $$\int_a^d f(x)\,dx$$ into $$\int_a^b f(x)\,dx + \int_b^c f(x)\,dx + \int_c^d f(x)\,dx$$?

Accepted Answer

Additivity over intervals.

Question 13

Which of the following is NOT a use of the definite integral?

Accepted Answer

Finding the slope of a function at a point.

Question 14

If $$f(x) is $$continuous and positive on $$[a, b]$$, what does $$\int_a^b f(x)\,dx$$ represent?

Accepted Answer

The total area between $$f(x)$$ and the x-axis from $$x=a to$$ $$x=b.$$

Question 15

For the function $$f(x) = 12 - x^2$ on $$[2, 4]$$, partitioned into four subintervals with midpoints c_k$, which formula correctly computes the Riemann sum S_4$?

Accepted Answer

$$S_4 = f(2.25$$) \cdot 0.5 + f(2.75) \cdot 0.5 + f(3.25) \cdot 0.5 + f(3.75) \cdot 0.5$$

Question 16

What is the main difference between left sums and right sums in approximating the area under a curve?

Accepted Answer

Left sums use the left endpoint of each subinterval for the rectangle height, while right sums use the right endpoint.

Question 17

If $$f(x) is $$positive and increasing on $$[a, b]$$, what can be said about the left sum $$L_n$$ and right sum $$R_n as$$ $$n 	o \infty$$?

Accepted Answer

Both $$L_n$$ and $$R_n$$ approach the same real number, which is the area under $$f(x)$$ from $$a to$$ $$b.$$

Question 18

Which scenario best illustrates the use of a definite integral in a real-world business context?

Accepted Answer

Calculating the total revenue generated by a continuous income stream over a period of time.

Chapter 5: Integration – The Definite Integral and Area Approximation

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