      Custom Search   PRACTICE PROBLEMS: 1. Find, by integration, the area under the curve y=x+4 bounded by the X axis and the lines x=2 and x=7 verify this by a geometric process. 2. Find the area under the curve bounded by the X axis and the lines x = 0 and x = 2 3. Find the area between the curve and the X axis, from x = -1 to x = 3 ANSWERS: SUMMARY The following are the major topics covered in this chapter: 1. Definition of integration: Integration is defined as the inverse of differentiation. where F(x) is the function whose derivative is the function f(x); is the integral sign; f(x) is the integrand; and dx is the differential. 2. Area under a curve: where (sigma) is the symbol for sum, n is the number of recŁtangles, is the area of each rectangle, and k is the designaŁtion number of each rectangle. Intermediate Value Theorem: where f(c) is the function at an intermediate point between a and b. where F(b) - F(a) are the integrals of the function of the curve at the values b and a. 3. Indefinite integrals: where C is called a constant of integration, a number which is independent of the variable of integration. Theorem 1. If two functions differ by a constant, they have the same derivative. Theorem 2. If two functions have the same derivative, their difference is a constant. 4. Rules for integration: The integral of a differential of a function is the function plus a constant. A constant may be moved across the integral sign. NOTE: A variable may NOT be moved across the integral sign. The integral of du may be obtained by adding 1 to the exŁponent and then dividing by this new exponent. NOTE: If n is minus 1, this rule is not valid. The integral of a sum is equal to the sum of the integrals. 5. Definite integrals: where b, the upper limit, and a, the lower limit, are given values. 6. Areas above and below a curve: If the graph of y = f(x), between x = a and x = b, has porŁtions above and portions below the X axis, then is the sum of the absolute values of the positive areas above the X axis and the negative areas below the X axis. ADDITIONAL PRACTICE PROBLEMS Evaluate the following integrals: ANSWERS TO ADDITIONAL PRACTICE PROBLEMS   