example, page 321 draw a graph of the signed area represented by the integral and compute it using...

Example, Page 321Draw a graph of the signed area represented by the integral and compute it using geometry.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

5 207. 25 x dx

Example, Page 321Draw a graph of the signed area represented by the integral and compute it using geometry.


1

110. 2x x dx


A function is integrable over [a, b] if all of the Riemann sums (not just the endpoint and midpoint approximations) approach one and the same limit L as the norm of the partition tends to zero, which wemay write as:.

0 0

1

lim , , limN

i iP Pi

L R f P C f c x

Assuming |R(f, P, C) – L| gets arbitrarily small as the norm||P|| tendsto zero, the limit is called the definite integral of f (x) over [a, b].


The definite integral is usually referred to as the integral of f over [a, b].The function f (x) inside the integral symbol is called the integrand.The numbers a and b are called the limits of integration.The independent variable in the function is used as the variableof integration.

Referring to b

af x dx


As illustrated in Figure 3, when f (x) is not positive for all x on [a, b],the definite integral yields the signed area between the graph and the x-axis. Signed area is defined as:

If the graph in Figure 3 represented the velocity of a particle, then theintegral from a to b would tell us the net displacement or movementof the particle from its starting point.


For the function illustrated in Figure 4.B. the Riemann sum converges to the signed area. In summary,


Calculate the definite integral of f (x) = 2x – 5 over [0, 3].


Evaluate the definite integralof f (x) = 3 – xover [0, 4].

Example, Page 321Use the basic properties of the integral and the formulas in the summary to calculate the integral.


3 2036. 6 7 1y y dy


Figure 8 illustrates the definiteintegral of f (x) = C over [a, b]for some C > 0. The integral may be evaluated by using Theorem 2.


Similar to limits, definite integrals have linearity properties as notedin Theorem 3.


If we reverse the limits of integration, we change the sign of the signed area yielded by the definite integral as noted in the following definition:

If the upper and lower limits of integration equal one another, the width of the interval is zero and


Show that for all values of b, (positive and negative):


Figure 9 more clearly illustrates the results obtained on the previous slide.


This theorem is illustrated in Figure 10.

Example, Page 321Calculate the integral, assuming


1 2 4

0 0 1

4

2

1, 4, 7

58.

f x dx f x dx f x dx

f x dx


Figure 11 illustrates Theorem 5, the Comparison Theorem.


If a function has a lower bound m and an upper bound M on [a, b],then the Comparison Theorem may be written algebraically as:

This is illustrated in Figure 12.


Using the information given in Figure 13, find the bounds of the definite integral of x–1 on [0.5, 2].

Example, Page 321Calculate the integral.


3

172. 2 4x dx

Homework

Homework Assignment #2 Read Section 5.3 Page 321, Exercises: 1 – 61(EOO), 65, 71


example, page 321 draw a graph of the signed area represented by the integral and compute it using...

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