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Chapter 5: Problem 30

Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. $$\int_{-1}^{3} \sqrt{4-(x-1)^{2}} d x$$

Short Answer

Expert verified
Question: Use geometry to evaluate the definite integral: \(\int_{-1}^{3} \sqrt{4-(x-1)^{2}} d x\) Answer: The value of the definite integral is \(2\pi\).

Step by step solution

01

Graph the integrand function

The given function is: $$f(x) = \sqrt{4-(x-1)^{2}}$$ This function represents a semi-circle with a radius of 2 and centered at (1, 0) on the x-axis. Now, let's sketch a graph of this integrand and find the region bound by the curve and the x-axis between the given integral bounds of \(x = -1\) and \(x = 3\).
02

Sketch the region

The definite integral represents the area under the curve of the function between the bounds, \(-1 \le x \le 3\). We'll sketch the graph and shade the region that represents our definite integral. On the graph, mark the points at x = -1 and x = 3. Draw a semi-circle between these bounds and the x-axis. Shade the region inside the semi-circle which represents the area of our definite integral.
03

Determine the geometric shape of the region

In our sketch, we can see that the shaded region represents a semi-circular shape. If we can find the area of this semi-circle, we would have calculated the definite integral using geometry.
04

Calculate the area of the semi-circle

A semi-circle has half the area of a circle. We know that the area of a circle is given by the formula: $$A = \pi r^2$$ Where \(A\) is the area and \(r\) is the radius. The radius of our semi-circle is 2. Therefore, the area of the semi-circle is: $$A_{semi-circle} = \frac{1}{2} (\pi (2)^2) = 2\pi$$
05

Evaluate the definite integral

Now that we've found the area of the semi-circular region, we can use geometry to evaluate the definite integral. The definite integral is equal to the area of the semi-circle: $$\int_{-1}^{3} \sqrt{4-(x-1)^{2}} d x = 2\pi$$ Thus, the value of the definite integral is \(2\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Semi-circle
In the context of our integral, a **semi-circle** is a two-dimensional geometric shape, representing half of a full circle. The given problem involves understanding the integrand function, which is \(f(x) = \sqrt{4-(x-1)^2}\). This function models a semi-circle because of its close relation to the equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\).
  • The center of our semi-circle is at \((1, 0)\).
  • The radius \(r\) is given by the constant term under the square root, so \(r = 2\).
  • This semi-circle is positioned on the x-axis stretching from \(x = -1\) to \(x = 3\).
Understanding the properties of the semi-circle helps us visualize the area for which we will later calculate using integrals.
Area Calculation
When calculating the **area** under a curve, we often use definite integrals. In this exercise, instead of using Riemann sums, we use the geometric property of the semi-circle. Here's how this works:
  • The area of a full circle is expressed by the formula \(A = \pi r^2\), where \(r\) is the radius.
  • A semi-circle is exactly half of a circle. Therefore, its area is \(A_{semi-circle} = \frac{1}{2} \pi r^2\).
  • For our semi-circle with \(r = 2\), this translates to:\(A_{semi-circle} = \frac{1}{2} \times \pi \times (2)^2 = 2\pi \).
By grasping that the area of a semi-circle calculates as half a circle's area, it becomes straightforward to understand that the definite integral of the given function over \([-1, 3]\) equals this value. This allows us to solve the problem by simple substitution into the geometric formula.
Geometric Interpretation
**Geometric interpretation** involves visualizing the function and region described by the definite integral. This interpretation is crucial for understanding the integrand function in a more tangible way.
  • In our problem, the curve described by the function \(f(x) = \sqrt{4-(x-1)^2}\) produces a semi-circle.
  • The limits of the integration, \(-1\) to \(3\), represent points on the x-axis that define the base of this semi-circle.
  • The problem asks us to interpret visually by sketching the curve on a graph, drawing the semi-circle, and shading the relevant area.
Through this graphical representation, one can "see" how the definite integral covers the area under the semi-circle, offering a clearer insight into what the calculation really represents鈥攁 measure of the area.
Function Graphing
**Function graphing** is a technique that helps to visualize mathematical concepts, which can be exceptionally helpful when dealing with integrals.
  • To graph the integrand function, which in this case is \(f(x) = \sqrt{4-(x-1)^2}\), we plot the semi-circle centered at \((1, 0)\) with a radius of 2.
  • The function limits, from \(x = -1\) to \(x = 3\), allow us to focus on the portion of the curve that forms the semi-circle.
  • Graphing involves marking the endpoints and drawing the semi-circle that touches these points and extends upward to reflect the value of the integrand function.
Employing graphing to visualize the function equips students with a non-abstract understanding of how the definite integral relates to actual geometric shapes. This method of visual analysis supports deeper engagement with the problem as a whole.

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