UNIVERSITY OF TECHNOLOGYMATERIAL ENGINEERING DEPT.

MATHEMATICSDr. Mohammed Ramidh

Semester I 2014/2015

Chapter .5

INTEGRATION

INTEGRATION

OVERVIEW:In this chapter we study a method to calculate the areas and volumes of these and other more general shapes. The method we develop, called integration, is a tool for calculating much more than areas and volumes.

”Indefinite Integrals and the Substitution Rule

the function ƒ is called the indefinite integral of ƒ with respect to x, and is symbolized by

The symbol ∫ is an integral sign. The function ƒ is the integrand of the integral, and x is the variable of integration.

EXAMPLE 1: Evaluate

Solution:

Basic Integration Formulas

˝The Power Rule in Integral Form

EXAMPLE 1: Using the Power Rule

EXAMPLE 2 : Adjusting the Integrand by a Constant

”The Substitution Rule

EXAMPLE 3: Using Substitution

EXAMPLE 4: Using Substitution

EXAMPLE 5: Using Identities and Substitution

EXAMPLE 6: Using Different Substitutions, Evaluate

EXERCISES 5.51. Evaluate the indefinite integrals in Exercises 1–12 by using the

given substitutions to reduce the integrals to standard form.

2. Evaluate the integrals in Exercises 13–48.

”The Definite Integral

Properties of Definite Integrals When ƒ and g are integrable on the interval [a , b], the definite integral satisfies Rules 1 to 7 in Table 5.3.

”Rules 2 through 7 have geometric interpretations,shown in Figure 5.11. the graphs in these figures are of positive functions, but the rules apply to general integrable functions.

EXAMPLE 1: Using the Rules for Definite Integrals, Suppose that.

EXAMPLE 3: Finding Bounds for an Integral, Show that the value of is less than 3\2

˝Area Under a Curve as a Definite Integral

In conclusion, we have the following rule for integrating f (x) = x.

This formula gives the area of a trapezoid

down to the line y = x (see Figure).

”The following results can also be established using

a Riemann sum calculation similar to that.

Solution

EXERCISES 5.3 1.

2.

3.

4.

5.

6.

7.

8.

9.

10. In Exercises, graph the function and find its average value over the given interval

EXERCISES 5.41. Evaluate the integrals in Exercises.

2. In Exercises, find the total area between the region and the x-axis.

3. Find the areas of the shaded regions in Exercises.

˝Area Between Curves

EXAMPLE 5: Changing the Integral to Match a Boundary Change

Solution

We subdivide the region at into subregionsA and B, shown in Figure we solve the equations

Only the value x = 4 satisfies the equation

˝Integration with Respect to y

If a region’s bounding curves are described by functions of y, the approximating

rectangles are horizontal instead of vertical and the basic formula has y in place of x. For regions like these.

EXAMPLE 6: Find the area of the region in Example 5 by integrating with respect

to y.

Solution :The region’s right-hand boundary is the line

˝Combining Integrals with Formulas from Geometry

The way to find an area may be to combine calculus and geometry.

EXAMPLE 7: Find the area of the region, shown in figure,

Solution:

EXERCISES 5.6:

1. Use the Substitution Formula to evaluate the integrals in Exercises.

2. Find the total areas of the shaded regions in Exercises

3. Find the areas of the regions enclosed by the lines and curves in Exercises .

4. Find the areas of the regions enclosed by the curves in Exercises.

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