Back59. Area of a segment of a circle
Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:
A_seg = (1/2) r² (θ - sin θ)
b. Find the area using calculus.
58–61. Arc length Find the length of the following curves.
y = x³/6 + 1/2x on [1,2]
The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2
a. Find the area of R
Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x² - y² = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as
x = cosh t = (eᵗ + e⁻ᵗ) / 2 and y = sinh t = (eᵗ - e⁻ᵗ) / 2.
b. In Chapter 8, the formula for the integral in part (a) is derived:
∫ √(z² − 1) dz = (z/2)√(z² − 1) − (1/2) ln|z + √(z² − 1)| + C.
Evaluate this integral on the interval [1, x], explain why the absolute value can be dropped, and combine the result with part (a) to show that:
t = ln(x + √(x² − 1)).
Area of a sector of a hyperbola: Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus
a. What is the area of R?
Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must satisfy xy < 1/2 in order for the area condition to be met. Then argue that the required probability is: 1/2 + ∫[1/2 to 1] (dx / 2x) and evaluate the integral.
A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.
a. Show that a satisfies the equation cosh 50/a − 1 = 10/a.
Area
In Exercises 11–14, find the total area of the region between the graph of ƒ and the x-axis.
ƒ(x) = x² - 4x + 3, 0 ≤ x ≤ 3
In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.
ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = x, y = 1/x², x = 2
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
√x + √y = 1, x = 0, y = 0
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
x = 2y², x = 0, y = 3
Find the extreme values of ƒ(x) = x³ - 3x², and find the area of the region enclosed by the graph of ƒ and the x-axis.
Find the total area of the region enclosed by the curve x = y²/³ and the lines x = y and y = -1.
Find the lengths of the curves in Exercises 19–22.
y = x¹/² ― (1/3) x³/² , 1 ≤ x ≤ 4