Let R be the region in the first quadrant that is bounded above by the polar curve

Question

r=4cosθ and below by the line θ=1, as shown in the figure above. What is the area of R ? 0.317 0.317 A 0.465 0.465 B 0.929 0.929 C 2.618 2.618 D 5.819

I am extremely confused with this question.

Answer 1

I do not know what the "figure above" looks like,

I do not know which aspects of the questions you find confusing, and

I am not able to attach a figure of my own.

So if you find my explanation even more confusing then let me know.

The curve r = 4cosθ consists of two circles of radius 2.

For clarity, let me call the radius of the circles s, i.e., s = 2.

One circle is centered around the point C = (s,0), the other around (-s,0).

Given that we care only about the first quadrant, only the first circle is relevant.

The line θ=1 goes through the origin, O, and has a slope of 1, which means

that it makes an angle of about 57° with the x-axis.

For clarity, let me call that angle α, i.e., α = 1, and the line is θ=α.

The line intersects the circle in some point P.

By definition of the curve r = 4cosθ, the length of the line segment OP is 4cosα.

The area to be found is the difference between the wedge OPC and the triangle OPC.

In the triangle OPC, let's calculate the angle at C.

OPC is an isosceles triangle with base OP and two branches OC and PC.

The angle at O is α, therefore the angle at P is also α,

and the angle at C is π-2α.

Knowing this angle allows us to calculate the area of the wedge OPC.

It is a portion of the area πs² of the whole circle.

The proportionality ration is the ration between π-2α and the whole angle 2π.

That is, the area of the wedge OPC is

πs²(π-2α)/2π = s²(π-2α)/2

To calculate the area of the triangle OPC we need its base OP, which has length 4cosα,

and we need its height; that is the distance of C from OP.

That distance is s×sinα.

Therefore the area of the triangle OPC is

4×cosα×s×sinα / 2

So the area to be determined is

s²(π-2α)/2 - 4×cosα×s×sinα/2

Substituting actual numbers

2²×(π-2)/2 - 4×cos(1)×2×sin(1)/2 = 0.465