Question

You found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves.. $$r=1+\sin \theta \text { and } r=1+\cos \theta$$

Short answer

Answer: The area of the region enclosed by both pairs of the curves is \(A = 4\pi - \frac{\pi}{2} \, units^2\).

Step-by-step solution

Find the intersection points of the polar curves

To find the intersection points of the polar curves \(r = 1 + \sin \theta\) and \(r = 1 + \cos \theta\), set them equal to each other and solve for \(\theta\): $$1 + \sin \theta = 1 + \cos \theta$$ Simplify and solve for \(\theta\): $$\sin \theta = \cos \theta$$ $$\tan \theta = 1$$ The solutions for \(\theta\) are \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\).

Set up the polar area integral

The area of a region, in polar coordinates, can be found using the formula: $$A = \frac{1}{2}\int_{\alpha}^{\beta} (f(\theta))^2 d\theta - \frac{1}{2}\int_{\alpha}^{\beta} (g(\theta))^2 d\theta$$ where \(f(\theta)\) and \(g(\theta)\) are the equations of the polar curves, and \(\alpha\) and \(\beta\) are the limits of integration, which are the intersection points found in Step 1. For this exercise, we have: \$f(\theta) = 1 + \sin \theta\$ \$g(\theta) = 1 + \cos \theta\$ \$ \alpha = \frac{\pi}{4}\$ \$ \beta = \frac{5\pi}{4}\$

Evaluate the integral

Now, we need to evaluate the integral to find the area enclosed by the curves: $$A = \frac{1}{2}\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} [(1+\sin \theta)^2 - (1+\cos \theta)^2] d\theta$$ First, expand the squares inside the integrand: $$A = \frac{1}{2}\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} (-2\sin\theta \cos\theta + 2\sin\theta - 2\cos\theta)d\theta$$ Now, we can integrate term by term and apply the integration limits: $$ \begin{aligned} A &= \frac{1}{2}[-\sin^2\theta + 2\sin \theta - 2\cos \theta\Big|_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \\ &= \frac{1}{2}[(1) - 2(8) \\ &= 4\pi - \frac{\pi}{2} \end{aligned} $$

Final Answer

The area of the region enclosed by both pairs of the curves is: $$A = 4\pi - \frac{\pi}{2} units^2$$

Key concepts

Intersection of Curves

Understanding the intersection of curves in polar coordinates is essential for solving various mathematical problems, such as finding the area enclosed by these curves. In polar coordinates, just like Cartesian coordinates, the intersection points are where the two equations are equal. Here, we have two polar curves given by:

• \( r = 1 + \sin \theta \)
• \( r = 1 + \cos \theta \)

To find the points where these curves intersect, we set the equations equal to each other:\[ 1 + \sin \theta = 1 + \cos \theta \]This simplifies to \( \sin \theta = \cos \theta \), which can be further simplified using the tangent identity \( \tan \theta = 1 \). Solving this equation gives us the angles \( \theta = \frac{\pi}{4} \) and \( \frac{5\pi}{4} \). These solutions are the angles at which the curves intersect in polar coordinates.This step gives us a critical piece of information: the limits of integration \( \alpha \) and \( \beta \) to use when finding the area shared by the intersecting curves.

Area in Polar Coordinates

The concept of area in polar coordinates might initially seem daunting, but it's a beautiful extension of familiar Cartesian techniques. In polar coordinates, the area of a region can be effectively calculated using integrals, leveraging the fascinating aspect of polar equations.For a curve \( r = f(\theta) \) in polar coordinates, the area \( A \) enclosed by the curve is determined by:\[ A = \frac{1}{2} \int_{\alpha}^{\beta} (f(\theta))^2 \, d\theta \]This formula derives from the concept that a small wedge of the circle can be confounded as a tiny segment, and these tiny segments sum to make a whole area.In our problem, we use this formula twice because we have two curves intersecting: one for radii \( f(\theta) = 1 + \sin \theta \) and another for radii \( g(\theta) = 1 + \cos \theta \).The task involves calculating the area enclosed simultaneously by both curves, which includes finding the integral from their intersection angles \( \alpha = \frac{\pi}{4} \) to \( \beta = \frac{5\pi}{4} \). The difference of these integrated functions over the defined angles provides the net area enclosed by the overlap of the two curves.This way of determining areas is elegant and taps into the nuances of polar symmetry and radial distance from the origin.

Integration Techniques

Integration is a fundamental technique in calculus used to find areas, and it's crucial in polar coordinates, especially when curves intersect. Here, simplifying expressions and ensuring correct limits is vital to solving the integral.In our problem:\[ A = \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \big((1+\sin \theta)^2 - (1+\cos \theta)^2\big) \, d\theta \]We simplify the expression, which involves expanding squares:

• \((1+\sin \theta)^2\) and \((1+\cos \theta)^2\)
• This simplifies to: \(-2\sin \theta \cos \theta + 2\sin \theta - 2\cos \theta\)

Next, integrate each term individually, making sure you correctly apply the limits of integration from \( \frac{\pi}{4} \) to \( \frac{5\pi}{4} \). The integration process can include trigonometric identities to simplify terms further.For the solution, each integrated term is evaluated at the bounds \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \), calculated to obtain a numerical value. The solution gives the area between curves, showcasing the power and elegance of integration in dealing with complex areas in polar coordinates.