Question

Find the area of the region that is bounded by the given curve and lies in the specified sector. r = e, 3 4 ≤ ≤ 3 2 A)3π B)2π C) π D)4π

Answer 1

The area of the region bounded by the given curve and lies in the specified sector is approximately 0.173 times π or 0.543π.

Step-by-step explanation:

The given equation is r = e. The area of a region bounded by a curve can be found using the formula for the area of a sector, which is A = 1/2 * r² * θ, where r is the radius of the sector and θ is the central angle in radians. In this case, the radius is given as e, which is approximately 2.71. The central angle can be found using the given bounds of 3π/4 to 3π/2. Subtracting the smaller angle from the larger angle gives 3π/2 - 3π/4 = π/4. Plug these values into the formula to find the area:

A = 1/2 * (2.71)² * (π/4) = 0.173π

Therefore, the area of the region bounded by the given curve and lies in the specified sector is approximately 0.173 times π or 0.543π.

Answer 2

The area of the region bounded by the curve r = e in the specified sector can be found using the polar coordinates area formula, resulting in A = ½ * e² * ½π.

Step-by-step explanation:

The area of a region bounded by a curve in polar coordinates is given by ½ ∫ r² dθ, where r is the radial coordinate and θ is the angular coordinate. In this specific question, the curve is defined by r = e (where e is the base of the natural logarithm, approximately equal to 2.71828) and the sector is between ¾π and ¾π + ½π, which corresponds to an angle of ½π radians or 90 degrees. Since r is constant, the area can be calculated simply by multiplying ½ the square of the radius by this angle: A = ½ * e² * ½π.