Q: Which of the following expressions represents the area under $$y = x^2$ from x = 0$ to x = 1$ as a limit of Riemann sums using right endpoints and n$ rectangles?

$$\lim_{n \to \infty} \sum_{i=1}^n \left( \frac{i}{n} \$$right)^2$$ \cdot \frac{1}{n}$$

Question 1

Suppose you want to estimate the area under the curve $$y = x^2$ from x = 0$ to x = 1$ using four rectangles and right endpoints. What is the sum of the areas of these rectangles?

Accepted Answer

$$0.46875$$

Question 2

Which of the following expressions represents the area under $$y = x^2$ from x = 0$ to x = 1$ as a limit of Riemann sums using right endpoints and n$ rectangles?

Accepted Answer

$$\lim_{n 	o \infty} \sum_{i=1}^n \left( \frac{i}{n} \$$right)^2$$ \cdot \frac{1}{n}$$

Question 3

Given the definite integral $$\$$int_0$$^1 (1 - x^2$) dx$$, what geometric region does this represent?

Accepted Answer

The area under the curve $$y = 1 - x^2$ from x = 0$ to x = 1$

Question 4

Which of the following is the correct formula for the area of a triangle with base $$b$$ and height $$h$$?

Accepted Answer

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

Question 5

If $$f(x) is $$continuous on $$[a, b]$$, which theorem guarantees that $$\int_a^b f(x) dx$$ exists?

Accepted Answer

Theorem 1: Integrability of Continuous Functions

Question 6

Which of the following is the correct sigma notation for the sum $$1 + 2 + 3 + ... + n$$?

Accepted Answer

$$\sum_{k=1}^n k$$

Question 7

A region $$R is $$bounded above by $$y = 1 - x^2$ and below by the x$-axis on $$[0, 1]$$. Which of the following best describes how to estimate the area of R$ using rectangles?

Accepted Answer

Divide $$[0, 1]$$ into $$n$$ subintervals, construct rectangles whose heights are the values of $$1 - x^2$ at chosen sample points, and sum their areas.

Question 8

Which property of definite integrals allows you to write $$\int_a^b f(x) dx = -\int_b^a f(x) dx$$?

Accepted Answer

Order of Integration

Question 9

Suppose $$f(x) = x^2$ and you use the left endpoint rule with n$ rectangles to estimate $$\$$int_0^1$$ $$x^2 dx$. Which of the following is the correct formula for the left Riemann sum?

Accepted Answer

$$\sum_{i=0}^{n-1} \left( \frac{i}{n} \$$right)^2$$ \cdot \frac{1}{n}$$

Question 10

Which of the following is the correct formula for the sum of the first $$n$$ squares?

Accepted Answer

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

Question 11

If $$f(x) \geq g(x)$$ for all $$x in$$ $$[a, b]$$, which property of definite integrals guarantees that $$\int_a^b f(x) dx \geq \int_a^b g(x) dx$$?

Accepted Answer

Domination

Question 12

A chemical reaction produces a concentration $$C(t) (in $$mol/L) over time $$t (in $$hours). What does $$\$$int_0$$^5 C(t) dt$$ represent in this context?

Accepted Answer

The total amount of substance produced over 5 hours

Question 13

Which of the following is the correct algebraic rule for finite sums: $$\sum_{k=1}^n (a_k + b_k)$$?

Accepted Answer

$$\sum_{k=1}^n a_k + \sum_{k=1}^n b_k$$

Question 14

If $$f(x) is a $$decreasing function on $$[0, 1]$$ and $$f(0) = 1$$, $$f(1) = e$$^{-1}$$, which of the following inequalities correctly estimates $$\$$int_0^1 f(x) dx$?

Accepted Answer

$$e^{-1}$$ \u003c \$$int_0^1 f(x) dx$$ \u003c 1$$

Question 15

Which of the following is the correct interpretation of the definite integral $$\int_a^b f(x) dx$$?

Accepted Answer

The net area between the graph of $$f(x)$$ and the $$x-$$axis from $$x = a to$$ $$x = b$$

Question 16

Given $$\$$int_0$$^8 f(x) dx = 12$$ and $$\$$int_0$$^{10} f(x) dx = 17$$, what is $$\$$int_8$$^{10} f(x) dx$$?

Accepted Answer

$$5$$

Question 17

Which of the following is the correct formula for the Riemann sum with equal-width subintervals for $$\int_a^b f(x) dx$$?

Accepted Answer

$$\lim_{n 	o \infty} \sum_{k=1}^n f\left(a + k \frac{b-a}{n}ight) \frac{b-a}{n}$$

Question 18

If $$f(x) is $$integrable on $$[a, b]$$ and $$k is a $$constant, which property of definite integrals allows you to write $$\int_a^b k f(x) dx = k \int_a^b f(x) dx$$?

Accepted Answer

Constant Multiple Rule

Area, Finite Sums, and the Definite Integral

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