Quiz about Riemann Sums and Integral Notation

Question 1

For each of the following integrals is $$\sum_{k=1}^n\ln\left(2+\frac{5}{n}k\right)\ \frac{5}{n}$$ the right Riemann sum aproximation with n subintervals of equal length?

Accepted Answer

$$\int_2^7\ln\left(x\right)dx$$

Answer

$$\int_2^7\ln\left(2+x\right)dx$$

Answer

$$\int_0^5\ln\left(2+5x\right)dx$$

Answer

$$\int_2^7\ln\left(2+5x\right)dx$$

Question 2

Which of the following is equivalent to the define integral $$\int_2^6\sqrt[]{x}dx?$$

Accepted Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{4}{n}\sqrt[]{2+\frac{4k}{n}}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{4}{n}\sqrt[]{\frac{4k}{n}}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{6}{n}\sqrt[]{\frac{6k}{n}}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{6}{n}\sqrt[]{2+\frac{6k}{n}}$$

Question 3

Which of the following definite integrals is equal to $$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{10k}{n}\left(\sqrt[]{1+\frac{5k}{n}}\right)\ \frac{5}{n}?$$

Accepted Answer

$$\int_0^52x\ \sqrt[]{1+x}dx$$

Answer

$$\int_1^{10}10\ \sqrt[]{x}dx$$

Answer

$$\int_1^62x\ \sqrt[]{x}dx$$

Answer

$$\int_0^510\ \sqrt[]{1+x}dx$$

Question 4

Which of the following definite integrals are equal to $$\lim_{n\rightarrow\infty}\sum_{k=1}^n\sin\left(-1+\frac{5k}{n}\right)\ \frac{5}{n}?$$

I. $$\int_{-1}^4\sin xdx$$

II. $$\int_0^5\sin\left(-1+x\right)dx$$

III. $$\int_0^1\sin\left(-1+5x\right)dx$$

Accepted Answer

I, II, and III

Answer

I only

Answer

II only

Answer

III only

Question 5

Which of the following is a left Riemann sum aproximation of $$\int_2^8\cos\left(x^2\right)dx$$ with n subintervals of equal length?

Accepted Answer

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

Answer

$$\sum_{k=1}^n\left(\cos\left(2+\frac{k-1}{n}\right)^2\right)\ \frac{1}{n}$$

Answer

$$\sum_{k=1}^n\left(\cos\left(\frac{6k}{n}\right)^2\right)\ \frac{6}{n}$$

Answer

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

Question 6

Which of the following definite integrals is equal to $$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{12k}{n}\cos\left(1+\frac{4k}{n}\right)\ \frac{4}{n}?$$

Accepted Answer

$$\int_0^43x\cos\left(1+x\right)dx$$

Answer

$$\int_1^512\cos xdx$$

Answer

$$\int_1^53x\cos xdx$$

Answer

$$\int_0^412\cos\left(1+x\right)dx$$

Question 7

Which of the following define integrals are equal to $$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(-2+\frac{8k}{n}\right)^3\ \frac{8}{n}?$$

I. $$\int_{-2}^6\ x^3dx$$

II. $$\int_0^8\left(-2+x\right)^3dx$$

III. $$\int_0^18\left(-2+8x\right)^3dx$$

Accepted Answer

I, II, and III

Answer

I only

Answer

II only

Answer

III only

Question 8

Which of the following is a left Riemann sum approximation of $$\int_1^7\left(4\ln x+2\right)dx$$ with n subintervals of equal length?

Accepted Answer

$$\sum_{k=1}^n\left(4\ln\left(1+\frac{6\left(k-1\right)}{n}\right)+2\right)\ \frac{6}{n}$$

Answer

$$\sum_{k=1}^n\left(4\ln\left(1+\frac{k-1}{n}\right)+2\right)\ \frac{1}{n}$$

Answer

$$\sum_{k=1}^n\left(4\ln\left(\frac{6k}{n}\right)+2\right)\ \frac{6}{n}$$

Answer

$$\sum_{k=1}^n\left(4\ln\left(1+\frac{6k}{n}\right)+2\right)\ \frac{6}{n}$$

Question 9

Which of the following limits is equal to $$\int_2^5x^2dx?$$

Accepted Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(2+\frac{3k}{n}\right)^2\ \frac{3}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(2+\frac{k}{n}\right)^2\ \frac{1}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(2+\frac{k}{n}\right)^2\ \frac{3}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(2+\frac{3k}{n}\right)^2\ \frac{1}{n}$$

Question 10

The function f is given by $$f\left(x\right)=\ln x.$$ The graph of f is shown. Which of the following limits is equal to the area of the shaded region?

Accepted Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\ln\left(1+\frac{3k}{n}\right)\ \frac{3}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(1+\ln\left(\frac{3k}{n}\right)\right)\ \frac{3}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\ln\left(\frac{4}{n}\right)\left(1+\frac{4k}{n}\right)$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\ln\left(1+\frac{4k}{n}\right)\ \frac{4}{n}$$

Question 11

Which of the following limits is equal to $$\int_0^3e^xdx?$$

Accepted Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n e^{\frac{3k}{n}}\ \frac{3}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n e^{\frac{k}{n}}\ \frac{1}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n e^{\frac{2k}{n}}\ \frac{2}{n}$$

Answer

$$\lim_{n\rightarrow\infty}\sum_{k=1}^n e^{\frac{4k}{n}}\ \frac{4}{n}$$

Question 12

Which of the following definite integrals is equal to $$\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{7k}{n}\sin\left(2+\frac{3k}{n}\right)\ \frac{3}{n}?$$

Accepted Answer

$$\int_0^3 7\sin\left(2+x\right)dx$$

Answer

$$\int_0^3 3x\sin\left(2+x\right)dx$$

Answer

$$\int_2^5 3x\sin xdx$$

Answer

$$\int_2^5 7\sin xdx$$

Question 13

Which of the following is a right Riemann sum approximation of $$\int_1^4\left(3x^2+5\right)dx$$ with n subintervals of equal length?

Accepted Answer

$$\sum_{k=1}^n\left(3\left(1+\frac{k}{n}\right)^2+5\right)\ \frac{3}{n}$$

Answer

$$\sum_{k=1}^n\left(3\left(1+\frac{k-1}{n}\right)^2+5\right)\ \frac{3}{n}$$

Answer

$$\sum_{k=1}^n\left(3\left(1+\frac{3k}{n}\right)^2+5\right)\ \frac{3}{n}$$

Answer

$$\sum_{k=1}^n\left(3\left(1+\frac{3\left(k-1\right)}{n}\right)^2+5\right)\ \frac{3}{n}$$

Riemann Sums and Integral Notation

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