Lecture 5

Faculty of Science; Department of Mathematics
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Readings For This Lecture

Keshet Course Notes:

Chapter 3, pages 43 to 50 (up to subsection 3.6)

Summary

Group 12: Add a summary of the lecture in this space. Include examples, discussion, and links to external sources, if desired.

Exercises

1. Consider the integral $\int _{0}^{a}\sin(x)dx$ . For what values of $a$ is this integral expression equal to zero?

Solution
The antiderivative of sin(x) is -cos(x). $\int _{0}^{a}-\cos(x)dx$ -cos(a)-(-cos(0)) = -cos (a) + 1 Therefore, cos(a) must equal 1. $a=2x\pi$ , where x is any integer.

2. Consider the integral $\int _{0}^{5}2^{x}-ax^{2}dx$ . For what values of $a$ is this integral expression equal to zero?

3. Discuss an application of integrals that makes explicit use of the 'negative area' property.

4. Consider the function $y=e^{x}$ on the interval $[0,1]$ . Find the area under this graph by using the Fundamental Theorem of Calculus.

5. Use an integral to approximate the sums $\sum _{k=1}^{N}{\sqrt {k}}$ .

6.Find the area between the two curves $y=1-x$ and y = x^{2}-1</math> for $x>0$ . Explain the relationship of your answer to the two integrals

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

7. Find the area under the graph of $y={\tfrac {1}{x}}$ from $x=1$ to $x=3$ . Extend the interval of integration to $[{\tfrac {1}{2}},3]$ and compute the area. How about $[{\tfrac {1}{4}},3]$ ? What happens as the left end point approaches 0? Can you explain?

8. Find the antiderivative of $x^{n}$ . Use this antiderivative to compute the integral $\int _{0}^{3}x^{n}dx$ . Confirm this formula by computing the area using sums of rectangular strips for the case $n=2$ .