Problem 68 - Essential Calculus

Question

Area, use a graphing utility to graph the region bounded by the graphs of the equations, and find the area of the region. $$ y=x \sin x, y=0, x=0, x=\pi $$

Step-by-Step Solution

Answer

The area of the region bounded by the given equations is \(\pi\) square units.

Step 1: Understanding and visualization

Firstly, visualize the given problem by plotting these equations on the graph. This will create a region bounded by these equations.

Step 2: Draw the regions

These equations create two lines and a curve that intersect at \(x = 0\) and \(x = \pi\). The line \(y=0\) is the x-axis, the line \(x=0\) is the y-axis, and \(x=\pi\) is a vertical line.

Step 3: Define integration limit

The region is located between \(x = 0\) and \(x = \pi\). These are also the lower and upper limits of the integration.

Step 4: Calculate the definite integral

Calculate the definite integral \(\int_0^\pi x \sin x \, dx \). The Fundamental Theorem of Calculus says that integration is the area under the curve. Thus, this integral represents the area of the region.

Step 5: Solve the definite integral

To solve this, we can use integration by parts with \( u = x \) and \( dv = \sin x \, dx \). The integral becomes: \[ \left.| -x \cos x \right|_0^\pi + \int_0^\pi \cos x \, dx \] which yields \(\pi\).

Key Concepts

Definite Integral

In calculus, the definite integral is a fundamental concept used to calculate the area under a curve within certain limits.

• A definite integral is written as \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the limits of integration.
• The function \(f(x)\) represents the curve, and \(dx\) signifies an infinitesimally small width along the x-axis.

By calculating the definite integral, we can determine the area between the curve and the x-axis from \(x = a\) to \(x = b\). This is exactly what we do while finding the area of a region bounded by curves on a graph. Breaking down the situation helps visualize the specific area under consideration, especially important when dealing with positive and negative areas that can cancel each other out.

Integration by Parts

Integration by parts is a technique used to solve integrals that are products of two functions. This method is especially useful when dealing with problems such as \( \int x \sin x \, dx \), which cannot be easily integrated using basic rules.The formula for integration by parts is derived from the product rule for differentiation:\[ \int u \, dv = uv - \int v \, du \]For the given problem, we choose:

• \(u = x\) and \(dv = \sin x \, dx\)
• Then, \(du = dx\) and \(v = -\cos x\)

Substituting these into the formula leads to the solution:\[ \int x \sin x \, dx = -x \cos x + \int \cos x \, dx \]This process simplifies the calculus, allowing us to solve integrals that at first glance seem complicated.

Graphing Utilities

Graphing utilities are software tools that help visualize mathematical equations by plotting them on a graph. This visualization is crucial for understanding the area between curves, as it allows us to see the regions we're calculating for the definite integral.By using a graphing utility:

• You can input equations like \(y = x \sin x\) and see how they plot on a coordinate graph.
• Adjusting the graph's scale and observing intersections makes it easier to determine the limits of integration.
• Seeing the plotted graph encourages a deeper grasp of how the integral represents the physical area.

Graphing utilities, therefore, not only provide a visual check but also serve as a great aid especially when solving complex integration problems.

Trigonometric Integration

Trigonometric integration is a method of integrating functions where trigonometric identities can be applied. These problems frequently involve integrals of sine, cosine, and other trigonometric functions.Trigonometric integration relies on various identities, such as:

• \( \int \sin x \, dx = -\cos x + C \)
• \( \int \cos x \, dx = \sin x + C \)
• Additionally, other identities such as \( \sin^2 x + \cos^2 x = 1 \) may be used to simplify expressions.

In this exercise, after integration by parts, we integrate the \( \cos x \) term separately, which highlights the role trigonometric identities play in simplifying and solving integrals. Understanding these principles fundamentally enhances your ability to deal with various trigonometric integrals, which are common in many advanced calculus problems.