[SOLVED] Vv156 Honors Calculus II Assignment 5

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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.^[1]

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.

• (1pt) y = x³, y = 8, x = 0; about y = 0.
• (1pt) x = 4y² − y³, x = 0; about y = 0.
• (1pt) y = x⁴, y = 0, x = 1; about x = 2.

Exercise 5.5 [Ste10, p. 453]

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

[1,3].

•  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]

• Use integration by parts to show that

Z                                        Z

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

•  If f and g are inverse functions and ^f′ is continuous, show that
•  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).
• 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 a_n sinnx. Show that the mth

n=1 coefficient a_m 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.

•  Verify that f is a probability density function.
• (Find.

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

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

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