MATH23MULTIVARIABLE CALCULUSIMPROPER INTEGRALS

GENERAL OBJECTIVEIdentify an Improper IntegralClassify Improper IntegralEvaluate improper integrals by applying limits in the definite integral Apply Improper Integrals in solving Area of Planes and Volume of Solids

At the end of the lesson the students are expected to:

IMPROPER INTEGRALDEFINITION:The definite integral

has the following assumptions:a. the limits of integration are finiteb. f(x) is continuous for every x in a < x < b otherwise the integral is an IMPROPER INTEGRAL

Types of Improper IntegralIntegral Over Infinite Intervals (Infinite Limits)

Integrals whose Integrals have Infinite Discontinuities (Discontinuous Integrand)

INFINITE LIMITSDefinite Integral with one or both limits that are infinite:

THEOREM

THEOREM

Evaluate the following integrals, if they convergeEXAMPLES

Determine if the following integrals converge or diverge. Evaluate the integrals, if they converge1.6.

2.7.

3.8.

4. 9.

5. 10.

EXERCISES

When f(x) is continuous for all values of x except in any point [a,b] or a < x < b, then DISCONTINUOUS INTEGRAND

THEOREM

THEOREM

GRAPHICAL INTERPRETATION

Determine if the following integrals converge or diverge. Evaluate the integrals, if they convergeEXAMPLES

Identify the points of discontinuity of the following functions. Determine if the following integrals converge or diverge. Evaluate the integrals, if they converge1. 7.

2.8.

3.9.

10.5.11.

6. 12.EXERCISES

Recall the formula for the plane Area and Volume of revolution as:

APPLICATIONS

Find the area bounded by the given curve and the x-axis:

and its asymptotes

(Hint: Multiply both numerator and denominator by ex).

EXAMPLES

Find the volume generated by revolving the area bounded by the given curve and the x-axis as it is revolved about the x-axis:

a. (Use Q1 area)

b. (Obtain the domain first of the function)

EXAMPLES

1) Determine whether the volume obtained by revolving about the x-axis the region bounded by the curve and the line x=e has an infinite volumeExercises 2. Let be the region bounded by the curve y= e-x and the x-axis x 0.(a) Find the Area of . (b) Find the Volume obtained by revolving about the x-axis.3. Let be the region bounded by the coordinate axes, thecurve y = and the line x=1. Find the area of

Examples for Volume Applications of Improper Integrals1.) Let be the region bounded by the curve Find the volume obtained by revolving about the x-axis.and the line x = 1.Answer: /3 cubic units.

2.) Find the volume of the solid generated by revolving around the y-axis the curveand the x-axis, x0.

Answer: 2 cubic units.

Examples:3.) Find the volume obtained by revolving about the y-axis the region bounded by the curve and the x-axis, 0

TEXTBOOKSAnton, Howard; Bivens Irl and Davis Stephen Calculus, Early Transcendentals, Chapter 7 pages 547 to 555Peterson, Thurman S Calculus With Analytic Geometry, Chapter 14 pages 289 to 292SUGGESTED READINGS

***Answer: (infinite volume)b)1 square meters cubic meters

(d) + ln(1 + )]

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