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SOLUTION This example differs from the previous ones because now we are evaluating a de nite integral (ie, an integral with numerical limits) We still use the integration by parts formula, keeping track of the numerical limits of integration We rst notice that, on the one hand, the integrand is not a product On the other hand, we certainly do not know an antiderivative for log x We remedy the situation by writing log x = 1 log x Now the only reasonable choice is to take u = log x and dv = 1 dx Therefore u(x) = log x v(x) = x and

Notice that the keys and values contained within the collection are available as separate lists through the Keys and Values properties IDictionary<TK, TV> defines the following indexer: TV this[TK key] { get; set; } You can use this indexer to get or set the value of an element You can also use it to add a new element to the collection Notice that the index is not actually an index, but rather the key of the item

IEnumerable<T> and IEnumerator<T> are the generic equivalents of the non-generic IEnumerable and IEnumerator interfaces described earlier They declare the same methods and properties, and work in the same way Of course, the generic versions operate on data of the type specified by the type argument IEnumerable<T> declares the GetEnumerator( ) method as shown here: IEnumerator<T> GetEnumerator( ) It returns an enumerator of type T for the collection Thus, it returns a type-safe enumerator

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7425 Computation of the Local Properties As shown above, the curvature and the pressure angle depend on the displacement program and its derivatives Thus, an adequate synthesis procedure of the displacement program is required The synthesis of the displacement program has been traditionally based on a limited set of functions giving rise to parabolic, harmonic, cycloidal, trapezoidal, and polynomial motions More complex motions, such as those appearing in indexing cam mechanisms, can be synthesized using spline functions, which are also known as nonparametric splines Nonparametric cubic splines, when applied to the displacement programs s(y) or f(y), take the simple forms

Method void Add(TK k, TV v)

Description Adds the key/value pair specified by k and v to the invoking collection An ArgumentException is thrown if k is already stored in the collection Returns true if the invoking collection contains k as a key Otherwise, returns false Removes the entry whose key equals k Attempts to retrieve the value associated with k, putting it into v Returns true if successful and false otherwise If k is not found, v is given its default value

We conclude this section by doing another de nite integral, but we use a slightly different approach from that in Example 73

(737)

bool ContainsKey(TK k) bool Remove(TK k) bool TryGetValue(TK k, out TV v)

Part II:

for i = 1, , n - 1, and with y representing s in the case of translating followers, f in the case of their oscillating counterparts Now, if we de ne the (n - 1)-dimensional vectors y, y , and y in exactly the same way as de ned in Subsec 741, then the counterparts of relations (729) and (730) take the forms Ay = 6Cy y = (F - GC)y (738) (739)

IEnumerator<T> has the same two methods as does the non-generic IEnumerator: MoveNext( ) and Reset( ) It also declares a generic version of the Current property, as shown here: T Current { get; } It returns a T reference to the next object Thus, the generic version of Current is type-safe There is one other difference between IEnumerator and IEnumerator<T>: IEnumerator<T> inherits the IDisposable interface, but IEnumerator does not IDisposable defines the Dispose( ) method, which is used to free unmanaged resources

SOLUTION We use integration by parts, but we apply the technique to the corresponding inde nite integral We let u = sin x and dv = cos x dx Then u(x) = sin x v(x) = sin x So sin x cos x dx = u dv v du sin x cos x dx du = u (x) dx = cos x dx dv = v (x) dx = cos x dx

supports the non-generic version of GetEnumerator( ) IEnumerator<T> also implements the non-generic IEnumerator interface, thus supporting the non-generic versions of Current

where A, C, F, and G are formally identical to their counterparts in Subsec 741 Their differences are that ai and bi in the entries of these matrices in that subsection change to Dyi and 1/Dyi, respectively In this case, obviously, Dy i y i +1 - y i

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