Problem 65 - Calculus for Biology and Medicine

Question

Use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-2}^{2}\left(\sqrt{4-x^{2}}-2\right) d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2\pi - 8 \).

Step 1: Understand the Function

The function given in the integral is \( f(x) = \sqrt{4-x^2} - 2 \). This can be broken into two parts: \( \sqrt{4-x^2} \) and \( -2 \). Recognize \( \sqrt{4-x^2} \) as the top half of a circle with radius 2 centered at the origin \((0,0)\).

Step 2: Identify the Graph Region

The graph described by \( \sqrt{4-x^2} \) is a semicircle from \( x = -2 \) to \( x = 2 \). From this shape, we subtract the constant line \( y = -2 \). This means we effectively have a semicircle of radius 2 centered at \((0,2)\) with a vertical shift down by 2 units.

Step 3: Calculate Area of Semicircle

The area of a full circle with radius 2 is \( \pi r^2 = \pi(2)^2 = 4\pi \). Since the circle is halved, the area under the semicircle is \( 2\pi \).

Step 4: Adjust for the Shift

The area we calculated represents the area from \(-2\) to \(2\) under \( \sqrt{4 - x^2} \). Subtract the area of the rectangle under \( -2 \). The rectangle has a height of 2 and a width of 4 (difference from \(-2\) to 2), so its area is \( 2 \times 4 = 8 \).

Step 5: Calculate the Signed Area

The integral \( \int_{-2}^{2} \left( \sqrt{4-x^2} - 2 \right) dx \) represents the area of the semicircle minus the area of the rectangle. So, the signed area is \( 2\pi - 8 \).

Step 6: Conclusion

Thus, the integral evaluates to the signed area under the graph, which is \( 2\pi - 8 \).

Key Concepts

Signed Area

When we talk about the signed area under a function, it's all about measuring the net space—considering whether sections of the graph are above or below the x-axis. This means that areas above the axis contribute positively, while those below take away from the total area.
In our exercise, we determined the total area by calculating the space between the semicircle and a shifted horizontal line. This process gives us both positive and negative contributions.
The result is a single value: \( 2\pi - 8 \), representing the net signed area of this combination of shapes. Calculating signed areas is crucial because it provides a fuller picture of how different parts of graph areas interact, rather than just their magnitude.

Semicircle

A semicircle is simply half of a circle. In this problem, we look at the semicircle defined by the equation \( \sqrt{4-x^2} \). This equation describes the top half of a circle centered at the origin with a radius of 2.
When visualizing this, imagine slicing a circle right through its center horizontally. The top portion is the semicircle that we consider.
The semicircle's area is half of the full circle's area. A full circle with radius 2 has area \( \pi \cdot (2^2) = 4\pi \). Therefore, the area under the top half is \( 2\pi \). Semicircles are common in calculus problems because their symmetric nature and simple area formula make computations more straightforward.

Geometry Interpretation

Geometry interpretation in calculus involves understanding shapes and surfaces described by equations. Here, \( \sqrt{4-x^2} - 2 \) visually represents two components: a semicircle and a horizontal line.
This process allows us to see how algebraic expressions translate into geometric figures.

• First, identify that \( \sqrt{4-x^2} \) is the semicircle from \(-2\) to \(2\).
• Next, recognize the shift caused by subtracting \(2\), moving our semicircle downwards by this amount.

A graph line pattern creates specific areas with positive or negative contributions. Understanding these shapes and movements is key to correctly interpreting and solving integration problems through geometry.

Definite Integral

The definite integral is a powerful calculus tool to compute the area under a curve, considering orientation through upper and lower bounds of the function. In this problem, we use the integral \( \int_{-2}^{2}\left(\sqrt{4-x^{2}}-2\right) d x \) to measure the area from \(-2\) to \(2\).
Calculating it involves evaluating how the area above the x-axis interacts with the area below due to the subtraction of \(2\). The definite integral combines these measurements, arriving at a single value reflecting this overall behavior.

• The area of any section above the x-axis adds to the integral's outcome.
• Subregions below subtract from the total, demonstrating the integral's ability to capture directionality.

Understanding definite integrals helps connect algebraic expressions with visual and spatial reasoning, key to solving problems involving areas.