Single Variable Calculus: Vol. 2, Early Transcendentals by James Stewart

By James Stewart

Good fortune on your calculus path starts off right here! James Stewart's CALCULUS: EARLY TRANSCENDENTALS texts are world-wide best-sellers for a cause: they're transparent, exact, and jam-packed with correct, real-world examples. With unmarried VARIABLE CALCULUS: EARLY TRANSCENDENTALS, 7th version, Stewart conveys not just the software of calculus that can assist you advance technical competence, but additionally provides an appreciation for the intrinsic fantastic thing about the topic. His sufferer examples and integrated studying aids can help you construct your mathematical self belief and accomplish your objectives within the path!

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Sn of equal width as in Figure 10. y y=ƒ S¡ 0 FIGURE 10 a S™ ⁄ S£ ¤ Si ‹ . . xi-1 Sn xi . . , ͓x nϪ1, x n ͔ where x 0 ෇ a and x n ෇ b. The right endpoints of the subintervals are x 1 ෇ a ϩ ⌬x, x 2 ෇ a ϩ 2 ⌬x, x 3 ෇ a ϩ 3 ⌬x, и и и Let’s approximate the ith strip Si by a rectangle with width ⌬x and height f ͑x i ͒, which is the value of f at the right endpoint (see Figure 11). Then the area of the ith rectangle is f ͑x i ͒ ⌬x . What we think of intuitively as the area of S is approximated by the sum of the areas of these rectangles, which is R n ෇ f ͑x 1 ͒ ⌬x ϩ f ͑x 2 ͒ ⌬x ϩ и и и ϩ f ͑x n ͒ ⌬x y Îx f(xi) 0 a ⁄ ¤ ‹ xi-1 b xi x FIGURE 11 Figure 12 shows this approximation for n ෇ 2, 4, 8, and 12.

Show that f is 26 3 ␲2 8 x sin x dx ഛ AREA FUNCTIONS of x iϪ1 and x i (that is, x *i ෇ sx iϪ1 x i ) and use the identity 1 1 1 ෇ Ϫ m͑m ϩ 1͒ m mϩ1 Խ f ͑x͒ Խ dx AREA FUNCTIONS 1. (a) Draw the line y ෇ 2t ϩ 1 and use geometry to find the area under this line, above the t-axis, and between the vertical lines t ෇ 1 and t ෇ 3. (b) If x Ͼ 1, let A͑x͒ be the area of the region that lies under the line y ෇ 2t ϩ 1 between t ෇ 1 and t ෇ x. Sketch this region and use geometry to find an expression for A͑x͒.

2 25. Let A be the area under the graph of an increasing contin- Rn Ϫ L n ෇ 28. Find the exact area of the region under the graph of y ෇ eϪx from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b). the curve y ෇ x 3 from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 ϩ 2 3 ϩ 3 3 ϩ и и и ϩ n 3 ෇ 27.

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