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.
Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2.
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region bounded by y = x²,y = 2x²−4x, and y = 0
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
Working with area functions Consider the function ƒ and its graph.
(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.
Determine the area of the shaded region in the following figures.
For the given regions R₁ and R₂, complete the following steps.
b. Find the area of region R₂ using geometry and the answer to part (a).
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.
Determine the area of the shaded region in the following figures.
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.
∫₀⁴ (8―2𝓍) d𝓍
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region in the first quadrant bounded by y = x/6 and y = 1−|x/2−1|
Find the area of the region described in the following exercises.
The region bounded by y=x^2−2x+1 and y=5x−9
Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.
ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]
Which of the following integrals correctly represents the area of the region enclosed by the curves and for ?
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = 2 sin x, y = sin 2x, 0 ≤ x ≤ π
Find the area enclosed by one loop of the curve .