A-Level CIE Further Maths 7.1 Polar Coordinates: Find the area of the region encl

Find the area of the region enclosed by the curve with polar equation $r = 2(1 + ext{cos} heta)$, for $0 ext{ } < heta < 2 ext{ } ext{pi}$. - CIE - A-Level Further Maths - Question 1 - 2013 - Paper 1

Question 1

Find the area of the region enclosed by the curve with polar equation $r = 2(1 + ext{cos} heta)$, for $0 ext{ } < heta < 2 ext{ } ext{pi}$.

Worked Solution & Example Answer:Find the area of the region enclosed by the curve with polar equation $r = 2(1 + ext{cos} heta)$, for $0 ext{ } < heta < 2 ext{ } ext{pi}$. - CIE - A-Level Further Maths - Question 1 - 2013 - Paper 1

Step 1

Use of the Area Formula

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Answer

To find the area enclosed by a polar curve, we use the formula:

$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$

In this problem, the equation is given as $r = 2(1 + \text{cos} \theta)$ , thus:

$r^2 = [2(1 + \text{cos} \theta)]^2 = 4(1 + \text{cos} \theta)^2$

Step 2

Setup the Integral

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Answer

Now we compute the definite integral from $0$ to $2\pi$ :

$A = \frac{1}{2} \int_{0}^{2\pi} 4(1 + \text{cos} \theta)^2 d\theta$

This can be simplified to:

$A = 2\int_{0}^{2\pi} (1 + 2\text{cos} \theta + \text{cos}^2 \theta) d\theta$

Step 3

Applying Double Angle Formula

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Answer

Using the double angle formula, we know that:

$\text{cos}^2 \theta = \frac{1 + \text{cos}(2\theta)}{2}$

Thus, the integral becomes:

$A = 2\int_{0}^{2\pi} \left(1 + 2\text{cos} \theta + \frac{1 + \text{cos}(2\theta)}{2}\right) d\theta$

Step 4

Integrate

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Answer

Now, we can simplify this further and integrate term by term:

$A = 2\int_{0}^{2\pi} \left(\frac{5}{2} + 2\text{cos} \theta + \frac{\text{cos}(2\theta)}{2}\right) d\theta$

This gives us three integrals to compute.

The integral of 1 is simply the limits multiplied by the constant.
The integral of $\text{cos} \theta$ over a full period ( $0$ to $2\pi$ ) is $0$ .
The integral of $\text{cos}(2\theta)$ over a full period ( $0$ to $2\pi$ ) is also $0$ .

Step 5

Final Calculation

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Answer

After performing all the integrations, we find that:

$A = 2 \left(\frac{5}{2} \cdot 2\pi\right) = 10\pi$

Therefore, the area enclosed by the curve is:

$A = 6\pi$

Accepting values, we find: $A \approx 18.8$

Find the area of the region enclosed by the curve with polar equation $r = 2(1 + ext{cos} heta)$, for $0 ext{ } < heta < 2 ext{ } ext{pi}$. - CIE - A-Level Further Maths - Question 1 - 2013 - Paper 1

Worked Solution & Example Answer:Find the area of the region enclosed by the curve with polar equation $r = 2(1 + ext{cos} heta)$, for $0 ext{ } < heta < 2 ext{ } ext{pi}$. - CIE - A-Level Further Maths - Question 1 - 2013 - Paper 1

Use of the Area Formula

Setup the Integral

Applying Double Angle Formula

Integrate

Final Calculation

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