Chapter 5 review    Recognize derivatives of famous functions, and know how to find integrals for sums (differences) of functions and constant multiples of functions. Integration with substitution. Never forget to express every term with the new variable in the original integral. Remember that in some tricky cases you need to use a little algebra to obtain this.

6.1    Area calculations for regions bounded by several functions.  Always start with a nice graph. If needed, split up the region into such subintervals that one function is larger than or equal to the other.  Remember that sometimes expressing all variablesin terms of y makes the problem easier.

6.2    Volume calculations by slicing in the direction perpendicular to the axis of rotation. Make sure that you (1) show  the details of your slicing with a picture and clear labeling of the variables, (2) express your calculation for the ith piece in your slicing, (3) set up the Riemann sum  and then  (4) express the definite integral.  Volume calculations for solids of revolution using disk and washer method. If the axis of revolution is not the x-axis or the y-axis, remember how to adjust the radii in the disk and washer methods. 

6.3    Volume calculations by slicing in the direction parallel to the axis of rotation. Make sure that you (1) show  the details of your slicing with a picture and clear labeling of the variables, (2) express your calculation for the ith shell in your slicing, (3) set up the Riemann sum  and then  (4) express the definite integral.  If the axis of revolution is not the x-axis or the y-axis, remember how to adjust the radii in the shell method.

6.4     Calculating work: Work = (weight of layer · distance moved).  Make sure you identify the pieces where the distance is approximately the same or the weigth is the same. Follow the same steps (1)-(4) as in all the other applications and remember that a correct picture and consistent use of your variables are especially crucial in these problems.  Be careful with the geometry;  when you work with sphere or hemisphere you will need the Pythagorian Theorem, when you work with cones or pyramids you will need to work with similar triangles.  Check your solution by working with the measurements thoroughly. 

6.5   Know how to find the average value of a function in an interval [a,b]. Understand the meaning of the Mean Value Theorem for integrals.

7.1   Integration by parts : always choose v dv  such that it is easy to find its antiderivative, and if you have a choice choose u such that its derivative simplifies the integration on the right hand side. Remember the examples when integration by parts is used twice.

7.2    If you have an odd power of sin(x) [or cos(x)] then rewrite the term  as sin(x) times an even power of sin(x) and then use the trig. identity  sin²(x) = 1 - cos²(x).  Next, use substitution  u = cos(x).  Otherwise use double angle formulas in your integral. Similar tehniques work for the functions tanx and sec x.

7.3    Understand the basic ideas of trigonometric substitution. Know when to choose and which function. Don't forget to restrict the domain to be able to work with the inverse functions. Always draw a nice picture of a right triangle to express the trigonomatric function needed in your solution.

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