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Lecture

Cal2Integration.pdf

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Department
Mathematics
Course Code
MAT1322
Professor
Termeh Kousha

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Integration and Its applications Professor: Termeh Kousha MAT 1322-Winter 2013 CONTENTS 2 Contents 1 Review of integration 3 1.1 Common Inde▯nite Integrals . . . . . . . . . . . . 3 1.1.1 Rules for Integrals . . . . . . . . . . . . 4 1.1.2 Integration by Substitution . . . . . . . . 5 1.1.3 Integration by Parts . . . . . . . . . . . .6 1.2 Finding the Area Between Two Curves . . . . . . 9 1.3 The Fundamental Theorem of Calculus . . . . . . 10 1.4 Area Between Intersecting Curves . . . . . . . . . 13 2 Improper Integrals 15 2.1 Improper Integrals of type I and II . . . . . . . . 15 2.2 The comparison Test . . . . . . . . . . . . . . . . 22 1 REVIEW OF INTEGRATION 3 1 Review of integration 0 Suppose that F(x) and f(x) are functions such that F (x) = f(x) then we say that F(x) is an antiderivative of f(x). For any antiderivative, F(x), of a function f(x), we use the notation Z f(x) dx = F(x) + C R where we say that f(x) dx is the inde▯nite integral (or gen- eral antiderivative) of f(x) with respect to x and we call C the constant of integration. 1.1 Common Inde▯nite Integrals 1. For any constant k, Z k dx = kx + C: 2. For any real number n except -1, Z n 1 n+1 x dx = x + C: n + 1 3. Z 1 dx = lnjxj + C: x 4. Z e dx = e + C: 1 REVIEW OF INTEGRATION 4 1.1.1 Rules for Integrals 1. For any constant k and function f(x), Z Z kf(x) dx = k f(x) dx: 2. For any functions f(x) and g(x), Z Z Z (f(x) + g(x)) dx = f(x) dx + g(x) dx: R Ex: Find each of the following inde▯nite integral 5e ▯3x +▯2 ▯1 7x dx. 0 3x Ex: Given that f (x) = e + 4 and f(0) = 2, ▯nd the function f(x). 1 REVIEW OF INTEGRATION 5 1.1.2 Integration by Substitution Integration by Substitution for Inde▯nite Integrals: 1. Choose u = g(x) such that you can see g (x) in the inte- grand. 2. Compute the di▯erential du = g (x) dx. 3. Substitute u and du into the integrand { there must not be any x or dx remaining. 4. Solve the resulting simpler integral. 5. Back-substitute u = g(x) into the ▯nal result. R We use integration by substitution for f(x) dx when f(x) is contains a function (g(x)) and its derivative (as a factor). Ex: Z 5 3x + 6 2 dx 2 x + 4x + 1 1 REVIEW OF INTEGRATION 6 1.1.3 Integration by Parts Integration by Parts Formula: For inde▯nite integrals: Z Z u dv dx = uv ▯ du v dx dx dx For de▯nite integrals: Z b dv Z bdu u dx = [uv]a▯ v dx a dx a dx Rules of Thumb: 1. Choose u to be a function that gets no more complicated when you take its derivative. dv 2. Choose dx to be a function that gets no more complicated when you take its integral. Ex: Find Z xe dx: 1 REVIEW OF INTEGRATION 7 Notes: ▯ It is crucial to make the right choice. Making the wrong 0 choice for u and v will usually give you a harder integral than the one you started with. For example, say you were asked to solve Z xe dx and chose u = ex v = x giving you x2 u = e x v = 2 Then integration by parts would give you Z 2 Z 2 xe = x e ▯ x e dx 2 2 | {z } even harder We ended up with a worse integral, so we know we made the wrong choice for u and v . ▯ Sometimes you have to do integration by parts twice. For example, say you were asked to solve Z x e dx The reasonable thing to do would be to choose 2 0 x u = x v = e giving you u = 2x v = ex 1 REVIEW OF INTEGRATION 8 Then integration by parts gives you Z Z x e dx = x e ▯ 2xe dx The second integral we can now do, but it also requires parts. We take u = 2x v = e x giving us 0 x u = 2 v = e So we have Z ▯ Z ▯ 2 x 2 x x x 2 x x x x e dx = x e ▯ 2xe ▯ 2e dx = x e ▯2xe +2e +C Rn general, you need to do n integration by parts to evaluate x e dx. ▯ In the case of de▯nite integrals, the integration by parts formula becomes Z Z b ▯ b uv dx = uv ▯ ▯ u vdx a a a Ex: Find Z ln(x) dx: 1 REVIEW OF INTEGRATION 9 1.2 Finding the Area Between Two Curves Suppose that you have two curves, y = f(x) and y = g(x) and some closed interval [a;b] such that ▯ f(x) and g(x) are continuous on [a;b] and ▯ f(x) ▯ g(x) on [a;b]. Picture: Question: How can we calculate the area between these two curves on the interval [a;b]? The area enclosed between two curves f(x) and g(x), where f(x) ▯ g(x) is given by Z b A = (f(x) ▯ g(x)) dx: a 1 REVIEW OF INTEGRATION 10 1.3 The Fundamental Theorem of Calculus For a function, f(x), which is continuous on the interval [a;b], we call Z b f(x) dx a the de▯nite integral of f(x) with respect to x from a to b. The numbers a and b are called the lower limit of integration and upper limit of integration respectively. Notice that, if f(x) ▯ 0 for all x in [a;b] then Z b Z b f(x) dx = f(x) ▯ 0 dx a a Rb so a f(x) dx is simply the area between the curve y = f(x) and the line y = 0 (the x-axis) from the vertical line x = a to the vertical line x = b. The Fundamental Theorem of Calculus: If f(x) is a contin- uous function on the interval [a;b] and F(x) is any antiderivative of f(x) t
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