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 $\int_{0}^{a} \sin (x) d x$ . For what values of $a$ is this integral expression equal to zero?

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

2. Consider the integral $\int_{0}^{5} 2^{x} - a x^{2} d x$ . 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

$I_{1} = \int_{0}^{1} (1 - x) d x$ and $I_{2} = \int_{0}^{1} (x^{2} - 1) d x$ .

7. Find the area under the graph of $y = \frac{1}{x}$ from $x = 1$ to $x = 3$ . Extend the interval of integration to $[\frac{1}{2}, 3]$ and compute the area. How about $[\frac{1}{4}, 3]$ ? 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 $\int_{0}^{3} x^{n} d x$ . Confirm this formula by computing the area using sums of rectangular strips for the case $n = 2$ .