Lecture 6

Faculty of Science; Department of Mathematics
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Readings For This Lecture

Keshet Course Notes:

Chapter 3, pages 50 to 60

Summary

Group 13: Add a summary of the lecture in this space. Include examples, discussion, and links to external sources, if desired.

Exercises

1. Use the fundamental theorem of calculus to compute the integral

$\int _{1}^{3}{\frac {2}{x}}dx$ .

Solution
The antiderivative of 2/x is 2ln(x). $\int _{1}^{3}{2}\ln(x)dx$ Apply FTC, 2ln(3)-2ln(1) =2ln(3) =2.19722...

2. Use the fundamental theorem of calculus to compute the integral of $x^{-2}$ over the interval $[-1,1]$ . Does your answer make sense, given that the graph of $x^{-2}$ is unbounded?

3. Use the fundamental theorem of calculus to compute the integral

$\int _{-1}^{1}2|x|dx$ .

4. Use the fundamental theorem of calculus to compute the integral

$\int _{a}^{T}{\frac {1}{x^{1/2}}}dx$ .

5. Use the fundamental theorem of calculus to compute the integral

$\int _{1}^{x}{\frac {1}{1+t^{2}}}dt$ .

6. Use the fundamental theorem of calculus to compute the integral

$\int _{0}^{x}\sin(3y)dy$ .

Solution
The antiderivative of sin(kx) = -1/k(cos(kx)). [ -1/3(cos(3x)] - [ -1/3(cos(3(0)))] = -1/3 (cos3x) + 1/3 (1) = -1/3 (cos3x - 1)

7. Use the fundamental theorem of calculus to compute the integral

$\int _{0}^{1}x(1-x)^{2}dx$ .

8. Use the fundamental theorem of calculus to compute the integral

$A(x)=\int _{0}^{x}{\sqrt {t}}-t^{2}$ .

Use the function $A(x)$ to find the maximum value of the integral.

9. If $f(x)$ is a continuous function on the interval $[a,b]$ , show that

$\left|\int _{a}^{b}f(x)dx\right|\leq \int _{a}^{b}\left|f(x)\right|dx$ .

10. Find the interval on which the function

$f(x)=\int _{0}^{x}{\frac {1}{1+t+t^{2}}}dt$

is concave upward.