Extra Problems for Chapter 7

11             An airplane travels in a straight line at 600 mph at an altitude of 4 miles. Find the rate of change of the angle of elevation one minute after the airplane passes directly over an observer on the ground.

12            A 40 ft ladder is to be propped up against a 15 ft wall as shown in the figure. What angle should the ladder make with the ground if the horizontal distance the ladder extends beyond the wall is to be a maximum?

24            Find the volume of the solid formed by rotating the region under the curve 0 ≤ x ≤ π, about the x-axis.

25            Find the volume of the solid generated by rotating the region under the curve y = tan x, 0 ≤ x ≤ π/4, about the x-axis.

In Problems 31-34, sketch the given function in (a) rectangular coordinates, (b) polar coordinates. Let 0 ≤ θ ≤ 2π.

35            Find the area of the polar region bounded by r = 1 + sin² θ.

36            Find the area of the polar region bounded by r - sin θ + cos θ.

37            Find the area of the polar region inside both the curves r = 1 - cos θ and r = 1 + cos θ.

38            Find the length in polar coordinates of the curve

r = sin⁴(¼θ), 0 ≤ θ ≤ π.

39            Find the surface area generated by rotating the polar curve

r = 1 - cos θ, 0 ≤ θ ≤ 7π/2, about the x-axis.

40            Use the Intermediate Value Theorem to prove that arctan y has domain (- ∞, ∞).

41            Use the Intermediate Value Theorem to prove that the domain of arcsec y is the set of all y such that y ≤ - 1 or y ≥ 1.

42            Prove that if/ is a differentiable function of x then

43            If u and v are differentiable functions of x then

44            Show that if f' and g are differentiable for all x then

45             Use integration by parts to prove the reduction formula

Hint:

46            Suppose y = f(x), a < x < b and x = g(y), c < y < d are inverse functions and are strictly increasing. Let y₀ = f(x₀). Prove that:

(a)    If f is continuous at x₀, g is continuous at y₀.

(b)    If f' (x₀) exists and f'(x₀) ≠ 0, then g'(y₀) exists.

47            Justify the following formula for the area of the polar region bounded by the continuous curves

θ = f(r), θ = g(r), a ≤ r ≤b, where 0 ≤ f(r) ≤ g(r) ≤ 2π.

48            Justify the following formula for the mass of an object in the polar region 0 ≤ r ≤ f(θ), a ≤ θ ≤b, with density ρ(θ) per unit area.

49            Justify the following formulas for the centroid of the polar region 0 ≤ r ≤ f(θ), a ≤ θ ≤ b.

Hint: The centroid of a triangle is located on a median ⅔ of the way from a vertex to the opposite side.

50            Find the centroid of the sector 0 ≤ r ≤ c, a ≤ θ ≤ b.

51             Find the centroid of the region bounded by the cardioid r = 1 + cos θ.