Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{0}^{2} \sqrt{1-(x-1)^{2}} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{\pi}{2}\).

Step 1: Recognize the Integral's Shape

The integral \(\int_{0}^{2} \sqrt{1-(x-1)^{2}} \, dx\) represents the area of a semicircle. This is because \(\sqrt{1-(x-1)^{2}}\) is the equation of a circle with radius 1, centered at (1, 0), but only the upper half-circle, due to the square root function.

Step 2: Identify the Semicircle

The equation \(\sqrt{1-(x-1)^{2}}\) is derived from the circle's equation \((x-1)^2 + y^2 = 1\). This is a circle with center at (1,0) and radius 1. The function describes the upper semicircle from \(x=0\) to \(x=2\).

Step 3: Calculate the Semicircle's Area

The full area of the circle with radius 1 is \(\pi r^2 = \pi \cdot 1^2 = \pi\). Therefore, the area of the upper semicircle is half of that, \((\frac{1}{2})\cdot\pi = \frac{\pi}{2}\).

Step 4: Evaluate the Integral using Geometry

Since the integral corresponds to the area of the upper semicircle from \(x=0\) to \(x=2\), which we found to be \frac{\pi}{2}\, this will be the value of the integral.

Key Concepts

Geometric Integration

Geometric integration is a fascinating approach that merges geometry with calculus. It's especially useful in calculating the area under curves that form recognizable geometric shapes, like circles or semicircles. In this context, geometric integration allows you to interpret an integral as an area without extensive calculation.
The process involves:

• Identifying the geometric shape represented by the integral.
• Deriving its dimensions, such as radius or center, using given equations.
• Calculating the area of the identified shape using geometric formulas, like the formula for the area of a circle.

This approach can be quicker and more intuitive, especially if you're familiar with the geometry involved, such as recognizing when an integral matches the equation of a circle or semicircle.

Semicircle Area

Understanding the area of a semicircle is essential in solving integrals involving semi-circular functions. A semicircle is simply half of a full circle. Its area is determined by halving the area of the corresponding full circle.
To calculate the area of a semicircle:

• Start with the full circle's area calculation, which is \(\pi r^2\), where \(r\) is the radius.
• Since a semicircle is half of a circle, its area is \(\frac{1}{2} \pi r^2\).

In the original exercise, a semicircle with a radius of 1 was used, so the area is \(\frac{\pi}{2}\). This showcases how integrating geometric shapes using known formulas can simplify integration.

Circle Equations

Circle equations in the Cartesian plane are a standard part of geometry, forming the basis for understanding more complex integration tasks. The general equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) represents the circle's center and \( r \) is its radius.
In our exercise, the integral involved a function \(\sqrt{1-(x-1)^{2}}\), which aligns with the equation of a circle centered at \( (1, 0) \) and a radius of 1. This is because \( (x-1)^2 + y^2 = 1 \) rearranges to show the relationship between x and y.
Using this understanding, you can quickly identify the shape (like a semicircle) described by integral bounds and expressions. Recognizing these patterns streamlines integration by relating it directly to the geometry of circles.