Chapter 3 Section 6

3.6 Review

3.6.1 Concepts

1. Write the definition of each of the following concepts:
   1. \(R(f,n)\); \(M(f,n)\); \(L(f,n)\).
   2. Riemann sum of \(f\) on \([a, b]\);
   3. definite integral of \(f\) on \([a, b]\)
   4. linearity of the definite integral.
2. Explain why \(\displaystyle \int_a^b f(x) \,dx =-\int_b^a f(x) \,dx\).

 

3. Show that if \(f\) and \(g\) are continuous functions such that \(f(x) \leq g(x)\) for all \(x\in [a.b]\), then \(\displaystyle \int_a^b f(x) \,dx \leq \int_a^b g(x) \,dx\).

 

4. Prove that (a) \(\displaystyle \sum_{i=1}^{n}i = \frac{n(n+1)}{2}\);    (b) \(\displaystyle \sum_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}\).

 

3.6.2 Exercises

 
In problems 1–4, for each function, find the exact value of \(R(f,n)\).

1. \(\displaystyle f(x)=\csc x \), \(\displaystyle \frac{\pi }{6}\leq x\leq \frac{5\pi }{6}\);    \(R(f,4)\).

 

2. \(\displaystyle f(x)=\cos^{-1}x \;\, x\in \left [-1, 1 \right]\);    \(R(f,4)\).

 

3. \(\displaystyle f(x)= e^{2x} \;\, x\in \left [\ln 2 , 4\ln 2\right ]\);    \(R(f,3)\).

 

4. \(\displaystyle f(x)= \ln x \;\, x\in \left [e^2 , 4e^2\right ]\);    \(R(f,3)\).

 

5. Find the exact value of \(M(f,3)\) for \(\displaystyle f(x)= 2x+3, \;\, x\in \left [-1,5\right ]\).

 
In problems 6 and 7, use right-endpoint approximations to evaluate the definite integral of \(f\) over the indicated interval.

6. \(f(x)= -3x^2+2x-1,\;\, x\in [-1,3]\).

 

7. \(f(x)= 2x^3-x-3,\;\, x\in [-1,1]\).

 
In problems 8 and 9, (a) sketch the graph of \(f\); (b) evaluate the definite integral over the interval by means of elementary geometry.

8. \(f(x)=|2x-3|-3, \;\, x\in [1,4]\).\(\vphantom{f(x)=\left \{
   \renewcommand{\arraystretch}{1.2}
   \begin{array}{cl}
   \sqrt{4-x^2}&-1\leq x \leq 0\\
   -x+2 & 0 < x\leq 4 \end{array} \right.}\)

 

9. \(f(x)=\left \{
   \renewcommand{\arraystretch}{1.2}
   \begin{array}{cl}
   \sqrt{4-x^2}&-1\leq x \leq 0\\
   -x+2 & 0< x \leq 4 \end{array} \right. \)

 

In problems 10 and 11, (a) find values \(m\) and \(M\) such that \(\displaystyle m \leq f(x) \leq M\) for all \(x\in \left [ a, b\right ]\);
 
(b) use part (a) to estimate \(\displaystyle I = \int_{a}^{b}f(x)\,dx\).

 

10. \(\displaystyle f(x) = \cos \sqrt{x},\;\, x\in \left [ \left (\frac{\pi}{6}\right )^2, \left (\frac{\pi}{2}\right )^2\right ]\).

 

11. \(\displaystyle f(x) = \tan^{-1} \sqrt x, \;\, x\in \left [\frac{1}{3}, 9 \right ]\).

 

12. Here is an interesting pattern:
    \[
    \renewcommand{\arraystretch}{1.25}
    \begin{array}{rcl}
    1 & = & 0 + 1\\
    2+3+4 & = & 1+8 \\
    5+6+7+8+9& = &8 + 27\\
    10 + 11+12+13+14+15+16 & = & 27+64
    \end{array}\]

    (a) Write the next row and verify the validity of each equality.

    (b) Write each row using sigma notation.

    (c) Conjecture what the 10-th row should be and verify your conjecture. Then write the row using sigma notation.

    (d) Write a general conjecture. Use sigma notation.

    (e) Prove your conjecture.