Vv156 Homework 5 -Solved

Exercise 5.1 [Ste10, p. 427] Sketch the region enclosed by the given curves and find its (unsigned) area.

           (i) y = cosx, y = 2 − cosx, 0 ≤ x ≤ 2π.                                              (ii) x = 2y2, x = 4 + y2.


Exercise 5.2 [Ste10, p. 427] Evaluate the integral and interpret it as the area of a region. Sketch the region.

Exercise 5.3 [Ste10, p. 440] Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.

Exercise 5.4 [Ste10, p. 445] Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

(a)  y = x3, y = 8, x = 0; about y = 0.

(b)  x = 4y2 − y3, x = 0; about y = 0.

(c)  y = x4, y = 0, x = 1; about x = 2.

Exercise 5.5 [Ste10, p. 453]

(a) If f is continuous and  , show that f takes on the value 4 at least once on the interval

[1,3].

(b)  Find the numbers b such that the average value of f(x) = 2+6x−3x2 on the interval [0,b] is equal to

Exercise 5.6 [Ste10, p. 470]

(a)  Use integration by parts to show that

                                                                                        Z                                        Z
                                                                                                 f(x)dx = xf(x) −             xf′(x)dx

(b) If f and g are inverse functions and f′ is continuous, show that

(c)  In the case where f and g are positive functions and b > a > 0, draw a diagram to give a geometric interpretation of part (b).

(d)  Use part (b) to evaluate 

Exercise 5.7 [Ste10, p. 478] Prove the formula, where m and n are positive integers.

N

Exercise 5.8 [Ste10, p. 478] A finite fourier sine series is given by the sum f(x) = X an sinnx. Show that the mth

n=1 coefficient am is given by                                                                            

Exercise 5.9 [Ste10, p. 528] Evaluate the integral

  Z ∞ dx (i)     √

              0               x(1 + x)

Exercise 5.10 [Ste10, p. 543] Find the exact length of the curve.

         (i) y = ln(secx), 1 ≤ x ≤ 2.                                                                .


p 2 with starting point Exercise 5.11 [Ste10, p. 544] Find the arc length function for the curve y = arcsinx + 1 − x

(0,1).

Exercise 5.12 [Ste10, p. 550] Find the exact area of the surface obtained by rotating the curve about the x-axis.

         (i) y = x3, 0 ≤ x ≤ 2.                                                                               (ii) 9x = y2 + 18, 2 ≤ x ≤ 6.

Exercise 5.13 [Ste10, p. 573] Let f(x) = 30x2(1 − x)2 for 0 ≤ x ≤ 1 and f(x) = 0 otherwise.

(a) Verify that f is a probability density function.

(b)  Find .

Exercise 5.14 [Ste10, p. 573] Let f(x) = c/(1 + x2).

(a) For what value of c is f a probability density function?

(b) For that value of c, find P(−1 < X < 1).