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SOLUTION Using the same philosophy about odd exponents as we did with sines and cosines, we substitute sec2 x 1 for tan2 x The result is tan x sec2 x 1 sec3 x dx We may regroup the terms in the integrand to obtain sec4 x sec2 x sec x tan x dx A u-substitution suggests itself: We let u = sec x and therefore du = sec x tan x dx Thus our integral becomes u4 u2 du = Resubstituting the value of u gives tan3 x sec3 x dx = EXAMPLE 715

// Implement the IComparable<T> interface public int CompareTo(Inventory obj) { return nameCompareTo(objname); } } class GenericIComparableDemo { static void Main() { List<Inventory> inv = new List<Inventory>(); // Add elements to the list invAdd(new Inventory("Pliers", 595, 3)); invAdd(new Inventory("Wrenches", 829, 2)); invAdd(new Inventory("Hammers", 350, 4)); invAdd(new Inventory("Drills", 1988, 8)); ConsoleWriteLine("Inventory list before sorting:"); foreach(Inventory i in inv) { ConsoleWriteLine(" " + i); } ConsoleWriteLine(); // Sort the list invSort();

On the other hand, the tangent of the pressure angle was derived in Angeles and LopezCajun (1991) as tan a (y ) = s (y ) - e s(y ) (779)

24:

ConsoleWriteLine("Inventory list after sorting:"); foreach(Inventory i in inv) { ConsoleWriteLine(" " + i); } } }

Now, substituting Eqs (777) and (778) into Eq (779) yields tan a (y ) = s (y ) - e s (y ) + c (780)

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Although implementing IComparable for classes that you create is often the easiest way to allow objects of those classes to be sorted, you can approach the problem in a different way by using IComparer To use IComparer, first create a class that implements IComparer, and then specify an object of that class when comparisons are required There are two versions of IComparer: generic and non-generic Although the way each is used is similar, there are some small differences, and each approach is examined here

The non-generic IComparer defines only one method, Compare( ), which is shown here: int Compare(object obj1, object obj2) Compare( ) compares obj1 to obj2 To sort in ascending order, your implementation must return zero if the objects are equal, a positive value if obj1 is greater than obj2, and a negative value if obj1 is less than obj2 You can sort in descending order by reversing the outcome of the comparison The method can throw an ArgumentException if the type of obj is not compatible for comparison with the invoking object An IComparer can be specified when constructing a SortedList, when calling ArrayListSort(IComparer), and at various other places throughout the collection classes The main advantage of using IComparer is that you can sort objects of classes that do not implement IComparable The following program reworks the non-generic inventory program so that it uses an IComparer to sort the inventory list It first creates a class called CompInv that implements IComparer and compares two Inventory objects An object of this class is then used in a call to Sort( ) to sort the inventory list

If, moreover, y1 and y2 are the values at which the pressure angle attains a maximum aM and a minimum am, the corresponding extremality conditions take the form s 1 (y ) - s 1 (y ) tan a M = 0 s 2 (y ) - s 2 (y ) tan a m = 0 (781a) (781b)

// Use IComparer using System; using SystemCollections; // Create an IComparer for Inventory objects class CompInv : IComparer { // Implement the IComparer interface public int Compare(object obj1, object obj2) { Inventory a, b; a = (Inventory) obj1; b = (Inventory) obj2; return anameCompareTo(bname); } }

Part II:

2

with si = s(yi), for i = 1, 2, and s (yi) and s (yi) de ned likewise The problem thus i i reduces to solving two independent nonlinear equations, (781a) and (781b), for the two unknowns y1 and y2 Furthermore, we have, from Eq (780), c tan a M + e = s 1 - s 1 tan a m c tan a M + e = s 2 - s 2 tan a m (782) (783)

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