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Homework Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398

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Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

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Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

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Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

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Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

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Homework, Page 398 49.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.

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Homework, Page 398 51.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.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 7: Techniques of Integration Section 7.1: Numerical Integration Jon Rogawski Calculus, ET First Edition

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The shaded area in Figure 1 cannot be calculated directly using a definite integral, since there is not an explicit antiderivative for Instead, we will rely on numerical approximation using the trapezoidal method

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we divide the interval [a, b] into N even intervals, the area may be found using the Trapezoidal Rule

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As 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 increasing the size of N, we can attain whatever degree of accuracy we may need.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 5 illustrates how a mid point estimate rectangle has the same area as a trapezoid where the top of the trapezoid is tangent to the curve at the midpoint of the interval.

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Example, Page 424 Calculate T N and M N for the value of N indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 424 Calculate T N and M N for the value of N indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 424 Calculate the approximation to the volume of the solid obtained by rotating the graph about the. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we assume f ″ (x) exists and is continuous on our interval, we may use Theorem 1.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 6 shows how trapezoidal estimates for areas under curves are more accurate for those with small values of f ″.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 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.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 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.

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Example, Page 424 State whether T N or M N overestimates or underestimates the integral and find a bound for the error. Do not calculate for T N or M N. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 424 Use the Error Bound to find a value of N for which the Error (T N ) ≤ 10 – 6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework Homework Assignment #16 Read Section 7.2 Page 424, Exercises: 1 – 11(Odd), 25, 29, 33, 37 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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