9. Graphical Applications of Integrals

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = ln x, for 1≤x≤4

For the given regions R₁ and R₂, complete the following steps.

a. Find the area of region R₁.

R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

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)).

Arc length:

Find the length of the curve y = x², 0 ≤ x ≤ √3/2.

135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).

Determine whether the following statements are true and give an explanation or counterexample.

a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

Find the lengths of the curves in Exercises 19–22.

y = x¹/² ― (1/3) x³/² , 1 ≤ x ≤ 4

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

 ∫₋₁² ( ―|𝓍| ) d𝓍

82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is

L = ∫(from 1 to 2)√(1 + 1/x²)dx.

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.

Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

Let ƒ(𝓍) = c, where c is a positive constant. Explain why an area function of ƒ is an increasing function.

Areas of regions Find the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval.

ƒ(𝓍) = sin 𝓍 on [―π/4, 3π/4]

Power lines 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.

c. Use your answer in part (b) to find a, and then compute the length of the power line.