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Trigonometric expressions arise frequently in our work, especially as a result of substitutions In this section we develop a few examples of trigonometric integrals The following trigonometric identities will be particularly useful for us I We have sin2 x = The reason is that cos 2x = cos2 x sin2 x = 1 sin2 x sin2 x = 1 2 sin2 x 1 cos 2x 2

HashSet<T> is a new collection added to the NET Framework by version 35 It supports a collection that implements a set It uses a hash table for storage HashSet<T> implements the ICollection<T>, IEnumerable, IEnumerable<T>, ISerializable, and IDeserializationCallback interfaces HashSet<T> implements a set in which all elements are unique In other words, duplicates are not allowed The order of the elements is not specified HashSet<T> defines a full complement of set operations, such as intersection, union, and symmetric difference This makes HashSet<T> the perfect choice for working with sets of objects HashSet<T> is a dynamic collection that grows as needed to accommodate the elements it must store Here are four commonly used constructors defined by HashSet<T>: public HashSet( ) public HashSet(IEnumerable<T> c) public HashSet(IEqualityCompare comp) public HashSet(IEnumerable<T> c, IEqualityCompare comp) The first form creates an empty set The second creates a set that contains the elements of the collection specified by c The third lets you specify the comparer The fourth creates a set that contains the elements in the collection specified by c and uses the comparer specified by comp There is also a fifth constructor that lets you initialize a set from serialized data In addition to the methods defined by the interfaces that it implements, HashSet<T> defines several of its own, most of which support various set operations The set operation methods defined by HashSet<T> are shown in Table 24-21 Notice that the arguments to these methods are IEnumerable<T> This means you can pass something other than another HashSet<T> as the second set Most often, however, both operands will be instances of HashSet<T> In addition to the properties defined by ICollection<T>, HashSet<T> adds Comparer, shown here: public IEqualityComparer<T> Comparer { get; } It obtains the comparer for the invoking hash set

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(771)

Part II:

Method public void ExceptWith(IEnumerable<T> set2) public void IntersectWith(IEnumerable<T> set2) public bool IsProperSubsetOf(IEnumerable<T> set2) public bool IsProperSupersetOf(IEnumerable<T> set2) public bool IsSubsetOf(IEnumerable<T> set2) public bool IsSuperSetOf(IEnumerable<T> set2) public bool SetEquals(IEnumerable<T> set2)

where an exponent k over a vector quantity indicates the kth power of the magnitude of the said vector, that is, r2 r r = ||r||2 Three surface integrals over Si are needed in the expression for wi, namely, A i = pp T dSi

II We have cos2 x = The reason is that cos 2x = cos2 x sin2 x = cos2 x 1 cos2 x = 2 cos2 x 1 Now we can turn to some examples 1 + cos 2x 2

Description Removes the elements in set2 from the invoking set Removes from the invoking set those elements not common to both the invoking set and set2 Returns true if the invoking set is a proper subset of set2 Returns true if the invoking set is a proper superset of set2 Returns true if the invoking set is a subset of set2 Returns true if the invoking set is a superset of set2 Returns true if the invoking set is equivalent to set2 This determination is independent of the order of the elements Changes the invoking set so that it contains all elements from both the invoking set and set2, except for those elements common to both sets This is usually called the symmetric difference of the two sets Adds the elements from set2 to the invoking set Duplicates are not included Thus, it creates a union of the two sets

public void SymmetricExceptWith(IEnumerable<T> set2)

(772a) (772b) (772c)

public void UnionWith(IEnumerable<T> set2)

SOLUTION Of course we will use formula II We write cos2 x dx = 1 + cos 2x dx 2 1 1 = dx + cos 2x dx 2 2 x 1 = + sin 2x + C 2 4

Here is an example that shows HashSet<T> in action:

Since p is a vector lying entirely in the plane Pi de ned by the polygon Si, it can be represented uniquely in the 2-D subspace as a 2-D vector of that plane Consequently, the GDT can be applied in this 2-D subspace to reduce the surface integrals (772a to c) to line integrals (Al-Daccak and Angeles, 1993), namely,

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