Question

Use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.\(r=\sin \theta+\cos \theta\) on the interval \(0 \leq \theta \leq \pi\)

Short answer

The area is \(\frac{\pi}{2}\).

Step-by-step solution

Convert Polar Equation to Cartesian Coordinates

To find the area covered by the curve given by the polar equation \(r = \sin\theta + \cos\theta\), we first convert this into Cartesian coordinates. Recall that in Cartesian coordinates, \(x = r \cos\theta\) and \(y = r \sin\theta\). Let's compute these:\[x = (\sin\theta + \cos\theta) \cos\theta = \cos\theta \sin\theta + \cos^2\theta\]\[y = (\sin\theta + \cos\theta) \sin\theta = \sin^2\theta + \sin\theta \cos\theta\]In polar form, this equation describes a circle since it simplifies to a form involving \(\sin\) and \(\cos\) which is typical for a geometric interpretation.

Simplify using Trigonometric Identity

Use the identity \(\sin\theta + \cos\theta = \sqrt{2}\sin(\theta + \frac{\pi}{4})\) for simplification. This form indicates that \(r = \sqrt{2}\sin(\theta + \frac{\pi}{4})\) which describes a circle or part of it. The response region will be a segment of this circular form.The transformation shows that \(r = \sin\theta + \cos\theta\) can be rewritten as a circle.

Compute Theoretical Area Using Geometry

In polar coordinates, the area enclosed by a curve described with \(r = a\sin(\theta + b)\) is given by half the product of circle area formula. Here, \(a = \sqrt{2}\), thus the complete circle area is \(\pi a^2 = 2\pi\). For half-circle (\(0\) to \(\pi\)), the area is \(\pi\).

Calculate Area by Definite Integral

Confirm the area by evaluating the definite integral for polar functions:\[A = \frac{1}{2} \int_{0}^{\pi} (\sin\theta + \cos\theta)^2 \, d\theta\]Split inside integral using identity \((a+b)^2 = a^2 + 2ab + b^2\):\[= \frac{1}{2} \int_{0}^{\pi} (\sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta) \, d\theta\]\[= \frac{1}{2} \int_{0}^{\pi} (1 + \sin 2\theta) \, d\theta\]Calculate integral:\(\frac{1}{2} \left[ \theta - \frac{1}{2}\cos 2\theta \right]_{0}^{\pi}\)= \(\frac{1}{2}[\pi - 0] = \frac{\pi}{2}\). Therefore, the polar region area is \(\pi/2\) per identity simplification.

Verify Consistency

Verify whether the definite integral result and geometric analysis are consistent and accurate. Carrying both calculation verifications ensures geometric depiction aligns with analytical computation.

Key concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to calculate the area under a curve within a specific interval. In real-world terms, it helps us find the exact area of a region bounded by curves. In this exercise, computing the area enclosed by the polar curve involves evaluating a definite integral for a function of variable \[\theta\].

For polar functions like \( r = \sin \theta + \cos \theta \), the definite integral is written in a form that accounts for the circular coordinate system. The area \(A\) can be determined using the formula:\[A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 \, d\theta\]where \( a \) and \( b \) are the boundary angles (here \(0\) to \( \pi \)). The function \([\sin\theta + \cos\theta]^2\) inside the integral indicates the region's complexity with alternating curves making up the boundaries. Splitting the integral and using trigonometric identities simplifies the computation further, ensuring an accurate calculation of area.

Trigonometric Identities

Trigonometric identities are mathematical tools to simplify expressions involving sine and cosine functions. They help in solving and simplifying equations, especially in complex forms. In this exercise, the identity \[sin\theta + \cos\theta = \sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)\]plays a crucial role.

This simplification transforms the polar equation, helping you visually interpret the equation as a circle in the polar coordinate system. Recognizing \(r = \sqrt{2}\sin(\theta + \frac{\pi}{4})\) eases the calculation as we deal with forms established by trigonometric identities.

• It improves understanding by converting a sum of sin and cos into a single sine term.
• Switching to the simpler circular identities aids in visualizing the geometry of the region.

These identities thereby support both theoretical analysis and calculation simplifications, reducing computational complexity.

Geometric Interpretation

Geometric interpretation allows one to visualize mathematical concepts in a way that connects step-by-step calculations with familiar shapes. By understanding that expressions like \( r = \sin \theta + \cos \theta \) can describe known geometrical entities, we can more easily grasp the scope of the problem.

Converting the polar coordinates into Cartesian or recognizing simpler geometric forms (like the circle in this example) allows you to see the problem beyond mathematical symbols.

• Imagine the curve \( r = \sin\theta + \cos\theta \) as a known circular segment when transformed, helping to intuitively understand the space it encloses.
• The visualization from trigonometric simplifications into a clearer geometric form ensures calculations align with visual expectations.

Thus, using geometry, you confirm the mathematical computations and gain a deeper appreciation of their accuracy and underlying simplicity.