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Final Review Guide

74 improper integrals improper integral of f over a

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7.4 Improper Integrals * Improper Integral of f over [ a, ): Let f be a continuous function on the unbounded interval [ a, ). Then, the imporper integral of f over [ a, ) is defined by Z a f ( x ) dx = lim b →∞ Z b a f ( x ) dx if the limit exists. * Improper Integral of f over ( -∞ , b ]: Let f be a continuous function on the unbounded interval ( -∞ , b ]. Then, the improper integral of f over ( -∞ , b ] is defined by Z b -∞ f ( x ) dx = lim a →-∞ Z b a f ( x ) dx if the limit exists. * Improper Integral of f over ( -∞ , ): Let f be a continuous function over teh unbounded interval ( -∞ , ). Let c be any real number and suppose both the improper integrals Z c -∞ f ( x ) dx and Z c f ( x ) dx 4
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are convergent. Then the improper integral of f over ( -∞ , ) is defined by Z -∞ f ( x ) dx = Z c -∞ f ( x ) dx + Z c f ( x ) dx. (It is usually helpful to choose c = 0) * Exercises: p.518 1-42 · Decide whether the following improper integrals converge or diverge. If they diverge, explain why. If they converge, compute what they converge to. 1. Z 0 -∞ xe x dx 2. Z π sin xdx 3. Z e 1 x ln x dx 4. Z π 2 cos x + x sin x x 2 7.5 Volumes of Solids of Revolution * Volume of a Solid of Revolution Formula: The volume V of the solid of revolution obtained by revolving the region below the graph of a nonnegative function y = f ( x ) from x = a to x = b about the x -axis is V = π Z b a [ f ( x )] 2 dx * The volume V of the solid of revolution obtained by revolving the region bounded above by the graph of the nonnegative function f ( x ) and below by the graph of the nonnegative function g ( x ), from x = a to x = b , about the x -axis is given by V = π Z b a ( [ f ( x )] 2 - [ g ( x )] 2 ) dx * Exercises: p.526 1-29 1. Calculate the volume of the solid obtained from rotating the following functions around the x -axis on the specified interval. (a) f ( x ) = 1 x on [0 , 1] (b) f ( x ) = 4 - x 2 on [0 , 2] (c) f ( x ) = e x on [0 , ln 3] 2. Calculate the volume of the solid obtained from rotating the area between two functions around the x -axis on the specified interval. (a) f ( x ) = xe x and g ( x ) = e x on [0 , 1] (b) f ( x ) = 1 and g ( x ) = x on [1 , 4] (c) f ( x ) = cos( x ) and g ( x ) = sin( x ) on [0 , π 4 ] 3. Calculate the volume of the solid obtained from rotating the area enclosed by the two functions around the x -axis. (a) f ( x ) = x 2 + 1 and g ( x ) = 2 x + 4 (b) f ( x ) = x 5 and g ( x ) = x 6 Chapter 8 Calculus of Several Variables 8.1 Functions of Several Variables * Terms: function of two variables, domain, independent variables, dependent variable, ordered triple, trace, level curve, contour map * Be able to compute the value of a function of two variables at a given point. * Be able to identify the domain of a function of two variables. 5
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* Be familiar with the 3D coordinate system. * Read pp. 535-536 for an explanation of level curves. * Exercises: p.538 1-24 8.2 Partial Derivatives * First Partial Derivatives of f ( x, y ): Suppose f ( x, y ) is a function of the two variables x and y . Then, the first partial derivative of f with respect to x at the point ( x, y ) is ∂f ∂x = lim h 0 f ( x + h, y ) - f ( x, y ) h provided the limit exists. The first partial derivative of f with respect to y at the point ( x, y ) is ∂f ∂y = lim k 0 f ( x, y + k ) - f ( x, y ) k provided the limit exists.
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