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

$\int_{1}^{3} \frac{2}{x} d x$ .

Solution
The antiderivative of 2/x is 2ln(x). $\int_{1}^{3} 2 \ln (x) d x$ 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

$\int_{- 1}^{1} 2 | x | d x$ .

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

$\int_{a}^{T} \frac{1}{x^{1 / 2}} d x$ .

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

$\int_{1}^{x} \frac{1}{1 + t^{2}} d t$ .

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

$\int_{0}^{x} \sin (3 y) d y$ .

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

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

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

$| \int_{a}^{b} f (x) d x | \leq \int_{a}^{b} | f (x) | d x$ .

10. Find the interval on which the function

$f (x) = \int_{0}^{x} \frac{1}{1 + t + t^{2}} d t$

is concave upward.