Find the area of the region bounded by the curve r=sinθ+cosθand lies in the sector 0≤θ≤π
Did you figure out the answer?
a = ∫[0,π] 1/2 (sinθ+cosθ)^2 dθ = (3π+2)/8 = 1.481
This seems reasonable, since the curve is the circle
(x - 1/2)^2 + (y - 1/2)^2 = 1/2
which has area π/2, but we have excluded a small segment subtending an arc of θ=π/2, with an area of 1/4 (π/2 - 1)
my bad. The area is ∫[0,3π/4]
To find the area of a region bounded by a curve in polar coordinates, we can use the following formula:
Area = (1/2) ∫[a, b] (r(θ))^2 dθ
In this case, the curve is given by r = sinθ + cosθ, and we are interested in the region within the sector 0 ≤ θ ≤ π.
To find the limits of integration (a and b), we need to determine the values of θ at which the curve intersects the boundaries of the sector.
First, let's find the intersection points. We have:
0 ≤ θ ≤ π
0 ≤ sinθ + cosθ
To find the values of θ that satisfy the above conditions, we can graph the curve r = sinθ + cosθ and shade the region of interest within the sector. We can use a graphing tool or software to do this.
By observing the graph, we can see that the curve intersects the boundaries of the sector at two points: θ = 0 and θ = π/2.
Therefore, the limits of integration are a = 0 and b = π/2.
Now, we can calculate the area using the formula:
Area = (1/2) ∫[0, π/2] (sinθ + cosθ)^2 dθ
The integral becomes:
Area = (1/2) ∫[0, π/2] (sin^2θ + 2sinθcosθ + cos^2θ) dθ
Expanding the terms, we get:
Area = (1/2) ∫[0, π/2] (1 + sin2θ + 2sinθcosθ + 1) dθ
Simplifying further, we have:
Area = (1/2) ∫[0, π/2] (2 + sin2θ + 2sinθcosθ) dθ
Now, we can integrate each term separately:
Area = (1/2) [2θ - (1/2)cos2θ + sin2θ] evaluated from θ = 0 to θ = π/2
Evaluating the limits, we get:
Area = (1/2) [2(π/2) - (1/2)cos(π) + sin(π)] - [(1/2)(0) - (1/2)cos(0) + sin(0)]
Simplifying further, we have:
Area = (1/2)(π - 0 + 0) - (0 - (1/2)(1) + 0)
Area = (1/2)(π - 1/2)
Finally, simplifying the expression, we get:
Area = π/4 - 1/8
Therefore, the area of the region bounded by the curve r = sinθ + cosθ and lies within the sector 0 ≤ θ ≤ π is π/4 - 1/8.