Question

Use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.\(r=6 \sin \theta+8 \cos \theta\) on the interval \(0 \leq \theta \leq \pi\)

Short answer

The area of the region defined by the polar equation \( r=6\sin\theta + 8\cos\theta \) on the interval \( 0 \leq \theta \leq \pi \) is confirmed using the definite integral.

Step-by-step solution

Identify the Formula for Area of Polar Curve

To find the area of the polar curve described by the equation, we can use the formula for the area of a polar region. The formula is \( A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta \), where \( r(\theta) \) is the polar equation and \( \alpha \) and \( \beta \) are the bounds of integration.

Substitute the Given Polar Equation

The polar equation given is \( r = 6 \sin \theta + 8 \cos \theta \). Substitute this into the area formula. Therefore, the formula becomes \( A = \frac{1}{2} \int_{0}^{\pi} (6 \sin \theta + 8 \cos \theta)^2 \, d\theta \).

Expand the Expression

Expand the expression \((6 \sin \theta + 8 \cos \theta)^2\). The expansion is \(36 \sin^2 \theta + 96 \sin \theta \cos \theta + 64 \cos^2 \theta \).

Apply Trigonometric Identities

Use trigonometric identities to simplify the terms: \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \) and \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \). Applying these identities allows you to rewrite \(36 \sin^2 \theta\) and \(64 \cos^2 \theta\) in terms of \(\cos 2\theta\).

Compute the Integral

Now compute the integral: \( \int_{0}^{\pi} [36\cdot \frac{1-\cos 2\theta}{2} + 96 \sin \theta \cos \theta + 64\cdot \frac{1+\cos 2\theta}{2}] \, d\theta \). Separate the integral, calculate each part separately, and then combine the results for the final area value.

Simplify and Solve the Integral

Carry out the integration step by step for each term. For terms involving \( \sin \theta \cos \theta\), use the identity: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Solve and simplify each integral to achieve the final area.

Confirm the Area with Calculator

After integrating and summing the values of the definite integrals, verify your solution by comparing it with a graphing calculator or reliable math software to ensure that the calculated area is correct.

Key concepts

Polar Coordinates

Polar coordinates are a way of describing a point in the plane using a distance from the origin and an angle, rather than the traditional Cartesian coordinates which use x and y values. Here, each point is determined by

• the radius, \( r \), which is the distance from the origin to the point,
• the angle, \( \theta \), measured from the positive x-axis.

The representation of curves with polar coordinates simplifies handling of equations involving cycles or curves emanating from the origin. In our problem, the curve is specified by the polar equation \( r = 6 \sin \theta + 8 \cos \theta \). This equation determines the radius \( r \) depending on the angle \( \theta \).Understanding polar coordinates makes it easier to calculate properties like the area since the integral setup directly corresponds to the curve's shape.

Definite Integral

A definite integral computes the total accumulation of a quantity, such as area under a curve, between two points on the x-axis. In calculus, the definite integral symbol is \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. It provides the net area between the curve and the x-axis over the specified interval.

In solving problems with polar coordinates, the definite integral measures the accumulated area within the curve described in polar form. For our exercise, the integration range is between \( 0 \) and \( \pi \), covering a specific section of the polar plot.

This approach allows efficient determination of areas where curves form intricate patterns, as typical in polar coordinate problems.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are particularly useful in simplifying complex expressions, especially in calculus problems.

In our exercise, identities are used for simplifying terms of the form \( \sin^2 \theta \) and \( \cos^2 \theta \) in the integral expression:

• \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \)
• \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \)

By applying these identities, the integral of \( (6 \sin \theta + 8 \cos \theta)^2 \) becomes manageable. This simplification is crucial to integrating the polar expression step by step effectively.Such identities are fundamental tools in calculus, enabling the transformation of complex integrals into simple terms that can be computed easily.

Area of Polar Regions

The area of polar regions is calculated using the formula \( A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta \). This formula is similar to the process of finding the area under a curve but adapted to the polar format.

Here, \( [r(\theta)]^2 \) represents the square of the radius function, while \( \alpha \) and \( \beta \) define the interval for the angle \( \theta \). This formula measures the sector area swept by the radius as \( \theta \) changes from \( \alpha \) to \( \beta \).

In the problem, \( r = 6 \sin \theta + 8 \cos \theta \), and utilizing this formula helps find precise areas bounded by these specific polar curves. Breaking down the complex expression with identities and integrating over the given bounds confirm the area's calculation's correctness. Understanding this process is essential for problems involving intricate polar-curved regions.