Pre-Calc Homework Solutions 300

Pre-Calc Homework Solutions 300 - 300 Section 7.3 1 : 2 46....

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Unformatted text preview: 300 Section 7.3 1 : 2 46. The curves are shown for m Quick Review 7.3 1. x 2 2. s 3. x 2 1 2 x2 r or 2 2 1 2 d 2 x2 or 2 8 , so Area s2 x2 . 2 [ 1.5, 1.5] by [ 1, 1] 4. In general, the intersection points are where which is where x 0 or else x 1 m x x2 1 mx, 5. b 6. b 7. b 8. x and h h h 2 3 x, so Area 1 bh 2 1 bh 2 1. Then, 1 bh 2 x2 . 2 x2 . 4 4 3 2 x . because of symmetry, the area is (1/m) 1 x, so Area x 2 2 x x2 1 0 mx dx 1 2 mx 2 (1/m) 1 0 , so Area 1 2 ln (x 2 2 1) 1 m ln 1 m m 1 m 1 2x 2x 1 ln (m) 1. s Section 7.3 Volumes (pp. 383394) b x x and h 1 bh 2 (2x)2 15 2 x . 4 Exploration 1 Volume by Cylindrical Shells 3xk xk2. Area 1 2 x 2 15 x, so 2 1. Its height is f (xk) 2. Unrolling the cylinder, the circumference becomes one dimension of a rectangle, and the height becomes the other. The thickness x is the third dimension of a slab with dimensions 2 (xk 1) by 3xk xk2 by x. The volume is obtained by multiplying the dimensions together. 3 9. This is a 3-4-5 right triangle. b Area 1 bh 2 4x, h 3x, and 6x . 2 3. The limit is the definite integral 0 2 (x 1)(3x x 2) dx. 10. The hexagon contains six equilateral triangles with sides of length x, so from Exercise 5, Area 6 4 3 2 x 3 3 2 x . 2 4. 45 2 Exploration 2 b Surface Area dy 2 dx dx Section 7.3 Exercises 1. In each case, the width of the cross section is w 2 1 x 2. (a) A r 2, where r s 2, where s s 2, where s w , so A(x) 2 w 2 2 1. a 2 y 1 The limit will exist if f and f are continuous on the interval [a, b]. (b) A 2. y b (1 x 2). 2(1 x 2). w, so A(x) w 2 w2 4(1 w 2 2 sin x, so 2 y 1 dy dx cos x and (c) A (d) A , so A(x) x 2). a dy 2 dx dx 4 3 2 w (see Quick Review Exercise 5), so 3 4 2 sin x 0 1 1 cos2 x dx 14.424. A(x) (2 1 x 2)2 3(1 x 2). 3. y 4 x, so 2 x 1 dy dx 2 and dx 36.177. 2 x 1 2 x 0 ...
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