9. Graphical Applications of Integrals

Which of the following integrals correctly represents the area of the region enclosed by the curves $y=x^2$ and $y=2x$ for $x\in [0,2]$?

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

Finding area

Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:

c. 2π ≤ x ≤ 3π.

41–48. Geometry problems Use a table of integrals to solve the following problems.

46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.

Find the area of the region described in the following exercises.

The region bounded by y=4x+4, y=6x+6, and x=4

In Exercises 139–142, find the length of each curve.

139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].

Centroid of a region

Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

41–48. Geometry problems Use a table of integrals to solve the following problems.

43. Find the length of the curve y = eˣ on the interval from 0 to ln 2.

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).

a. Write a single integral that gives the area of R.

Determine the area of the shaded region in the following figures.

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.

Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x=b.

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

c. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b).

Find the area enclosed by one loop of the polar curve $r=4\sin \left(3\theta \right)$.