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HomeworkHomework Assignment #47Read Section 7.1Page 398, Exercises: 23 51(Odd)Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.
Homework, Page 39849.Sketch the hypocycloid x 2/3 + y 2/3 = 1 and find the volume of the solid obtained by revolving it about the x-axis.
Homework, Page 39851.A bead is formed by removing a cylinder of radius r from the center of a sphere of radius R. (Figure 12) Find the volume of the bead with r = 1 and R = 2.
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyChapter 7: Techniques of IntegrationSection 7.1: Numerical IntegrationJon Rogawski Calculus, ET First Edition
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyThe shaded area in Figure 1 cannot be calculated directly using a definite integral, since there is not an explicit antiderivative forInstead, we will rely on numerical approximation using the trapezoidal method
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyIf we divide the interval [a, b] into N even intervals, the area may be found using the Trapezoidal Rule
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyAs shown in Figure 3, the area of the trapezoidal segment is equal to the average of the left- and right-RAM areas. As shown in table one, by increasingthe size of N, we can attain whateverdegree of accuracy we may need.
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyFigure 5 illustrates how a mid point estimate rectangle has the samearea as a trapezoid where the top of the trapezoid is tangent to the curve at the midpoint of the interval.
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Example, Page 424Calculate TN and MN for the value of N indicated.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Example, Page 424Calculate TN and MN for the value of N indicated.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Example, Page 424Calculate the approximation to the volume of the solid obtained by rotating the graph about the .Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyIf we assume f (x) exists and is continuous on our interval, we may use Theorem 1.
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyFigure 6 shows how trapezoidal estimates for areas under curvesare more accurate for those with small values of f .
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyFigure 6 shows the points we would use in calculating T6 and M6 for an approximation to the area of the shaded region in Figure 8.
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and CompanyFigure 10 illustrates how trapezoids provide an underestimate of areas under concave down curves and midpoints provide over-estimates. The opposite holds true for concave up curves.
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Example, Page 424State whether TN or MN overestimates or underestimates the integral and find a bound for the error. Do not calculate for TN or MN.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Example, Page 424Use the Error Bound to find a value of N for which the Error (TN) 10 6.Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
HomeworkHomework Assignment #16Read Section 7.2Page 424, Exercises: 1 11(Odd), 25, 29, 33, 37Rogawski CalculusCopyright 2008 W. H. Freeman and Company
Rogawski CalculusCopyright 2008 W. H. Freeman and Company
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