1322Notes(4) - MAT1322 Notes By Dr. Hua Contents Chapter 5...

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MAT1322 – Notes — By Dr. HuaContentsChapter 5 – Integrals35.10 Improper Integrals. . . . . . . . . . . . . . . . . . . . . . .3Chapter 6 – Applications of Integration76.1 More about Areas. . . . . . . . . . . . . . . . . . . . . . . . .76.2 Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86.3 Volumes by Cylindrical Shells. . . . . . . . . . . . . . .96.4 Arc Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116.5 Average Value of a Function. . . . . . . . . . . . . . . .126.6 Applications to Physics and Engineering. . . . .12Chapter 7 – Differential Equations187.1 Modeling with differential equations. . . . . . . .187.2 Direction Fields and Euler’s method. . . . . . . . .187.3 Separation of variables. . . . . . . . . . . . . . . . . . . . .217.4 Exponential Growth and Decay. . . . . . . . . . . . .247.5 The Logistic Equation. . . . . . . . . . . . . . . . . . . . . .27Chapter 8 – Infinite Sequences and Series328.1 Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .328.2 Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338.3 The Integral and Comparison Test; Estimatingsums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .348.4 Other Convergence Tests. . . . . . . . . . . . . . . . . . .378.5 Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .398.6 Representations of Functions as Power Series.428.7 Taylor and Maclaurin Series. . . . . . . . . . . . . . . .441
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Chapter 9 – Vectors and the Geometry of Space499.6 Functions and Surfaces. . . . . . . . . . . . . . . . . . . . .49Chapter 11 – Partial Derivatives5011.1 Functions of Several Variables. . . . . . . . . . . . . .5011.3 Partial Derivatives. . . . . . . . . . . . . . . . . . . . . . . .5111.4 Tangent Planes and Linear Approximations.5311.5 The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . .5511.6 Directional Derivatives and the Gradient Vec-tor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .572
Chapter 5 – Integrals5.10 Improper IntegralsType I: Infinite Intervalsaf(x)dx=limt→∞taf(x)dx,b-∞f(x)dx= limt→∞btf(x)dx,-∞f(x)dx=cf(x)dx+c-∞f(x)dx.Definition: Integral is convergent (or divergent)Integral is a finite number (or).Example 111x2dx= 1.Example 211xdx=.Example 31-∞11 +x2dx=3π4.Example 4-∞e-|x|dx= 2.Example 5Determine if the integralI=2xe-xdxis convergent or divergent and evaluate if it is convergent.3
Solution.I= limt→∞t2xe-xdx(integration by parts: letu=xanddv=e-xdx)= limt→∞x(-e-x)|t2-t2-e-xdx= limt→∞(-xe-x-e-x)|t2= limt→∞-(x+ 1)e-x|t2= limt→∞(-(t+ 1)e-t+ 3e-2)= 3e-2(where limt→∞(t+ 1)e-t= 0 by L’Hospital’s Rule).Type 2: Discontinuous IntegrandsIff(x) is continuous on [a, b), thenbaf(x)dx= limtbtaf(x)dx;Iff(x) is continuous on (a, b], thenbaf(x)dx= limtabaf(x)dx;Iff(x) isdiscontinuousatc:a < c < b, thenbaf(x)dx=caf(x)dx+bcf(x)dx.Example 63213-xdx= limt3t213-xdx= limt3[-23-x]t2= 2.Example 7Determine if the integral201x-1dxis convergent or divergent and evaluate if it is convergent.Example 8e0lnxdx= 0.4
p-IntegralExample 911xpdx=1p-1,ifp >1;divergent,ifp1.Example 10101xpdx=11-p,ifp <1;divergent,ifp1.Comparison Test for Improper IntegralIff(x) andg(x) are continuous andf(x)g(x)0 onxa. Then(i)af(x)dxis convergent =ag(x)dxis convergent;(ii)ag(x)dxis divergent =af(x)dxis divergent.Example 1111x3+ 1dx=convergent.1x3+ 11x3/2.Example 1281 +xx-6dx=divergent.

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Term
Winter
Professor
Kousha
Tags
Calculus, Improper Integrals, Integrals, dx, lim P, example
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Larson
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