How do you find the area of the region bounded by the polar curve #r=3cos(theta)# ?

Answer 1

The area of the region is #9/4pi#.

Let us look at some details.

The region is a disk, which looks like this:

If you are allowed to use the formula for the area of a circle, then

#A=pir^2=pi(3/2)^2=9/4pi#

If you wish to use integration, then

#A=int_0^{pi}\int_0^{3cos theta}rdrd theta#

#=int_0^{pi}[r^2/2]_0^{3cos theta}d theta#

#=int_0^{pi}{9cos^2theta}/2 d theta#

by the trig identity: #cos^2theta=1/2(1+cos2theta)#,

#=9/4int_0^{pi}(1+cos2theta) d theta#

#=9/4[theta+{sin2theta}/2]_0^{pi}#

#=9/4pi#

I hope that this was helpful.

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Answer 2

To find the area of the region bounded by the polar curve r = 3cos(θ):

  1. Determine the interval of θ where the curve intersects to form the desired region.
  2. Set up the integral for the area using the formula for the area enclosed by a polar curve.
  3. Integrate the expression over the determined interval of θ.

For the curve r = 3cos(θ), it forms a complete loop as θ varies from 0 to π. This loop creates the enclosed region.

The formula for the area enclosed by a polar curve is:

A = (1/2)∫[a, b] r(θ)^2 dθ

where a and b are the limits of integration, and r(θ) is the polar curve function.

Substituting r(θ) = 3cos(θ) into the formula, we have:

A = (1/2)∫[0, π] (3cos(θ))^2 dθ

A = (9/2)∫[0, π] cos^2(θ) dθ

Using the identity cos^2(θ) = (1 + cos(2θ))/2, the integral becomes:

A = (9/2)∫[0, π] (1 + cos(2θ))/2 dθ

A = (9/4)∫[0, π] (1 + cos(2θ)) dθ

Now integrate term by term:

A = (9/4) [θ + (1/2)sin(2θ)] | [0, π]

Evaluate the integral:

A = (9/4) [π + (1/2)sin(2π) - 0 - (1/2)sin(0)]

Since sin(2π) = sin(0) = 0, this simplifies to:

A = (9/4)π

So, the area of the region bounded by the polar curve r = 3cos(θ) is (9/4)π square units.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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