Problem 41 - Calculus for Biology and Medicine

Question

$$ \int_{-2}^{2}\left(\sqrt{4-x^{2}}-2\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify Symmetry

Notice that the integrand, \( f(x) = \sqrt{4-x^2} - 2 \), is an even function because \( f(-x) = f(x) \). This is demonstrated by the fact that both \( \sqrt{4-x^2} \) and \(-2\) are even components.

Step 2: Integrate the Square Root Function

Recall the integral of \( \sqrt{4-x^2} \), which represents a semicircle of radius 2. The integral from \(-2\) to 2 of \( \sqrt{4-x^2} \) calculates the area of the semicircle, which is \( \frac{1}{2} \pi \times 2^2 = 2\pi \).

Step 3: Integrate the Constant

The integral of the constant \(-2\) over \([-2, 2]\) is \(-2 \times (2 - (-2)) = -2 \times 4 = -8\).

Step 4: Combine the Integrals

Now combine the results from the previous steps: \( 2\pi - 8 \). This uses the property \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \).

Step 5: Calculate the Final Result

Substitute the values calculated: \( 2\pi - 8 \) gives the final result of the integral.

Key Concepts

Definite Integrals

A definite integral is a fundamental concept in calculus. It represents the accumulation of quantities, such as areas under curves, between two specific limits. When you see notation like \( \int_{a}^{b} f(x) \, dx \), it's asking for the integral of the function \( f(x) \) from \( x = a \) to \( x = b \). This area can be positive, negative, or zero, depending on whether the curve is above or below the x-axis.

To calculate a definite integral, follow these steps:

• Evaluate the antiderivative (indefinite integral) of the function over the interval.
• Subtract the value of the antiderivative at the lower limit (a) from its value at the upper limit (b).

The result is the net "area" between the curve and the x-axis over \([a, b]\). In the context of the provided example, we are integrating the function \( f(x) = \sqrt{4-x^2} - 2 \) over the interval from \(-2\) to \(2\). This involves calculating the areas described by the function.

Even Function

Even functions have a symmetrical property where \( f(-x) = f(x) \). This means the graph of an even function mirrors itself across the y-axis. Recognizing an even function is useful when computing definite integrals over symmetric intervals like \([-a, a]\). For such intervals, the integral of an even function can often be simplified.

In the example, determining that \( f(x) = \sqrt{4-x^2} - 2 \) is even helps simplify the integration process. Because the interval is symmetric and \( f(x) \) is even, the positive and negative parts balance each other out. Instead of working this out for one half and then doubling, both halves yield the same contribution. This symmetry property is a huge help to mathematicians and students alike when it comes to calculating areas under curves.

Area Under a Curve

Finding the area under a curve is a common use for definite integrals. This method lets us calculate the "net" area, which considers both the parts of the curve above and below the x-axis. The notion applies to various applications such as physics, economics, and more, where calculating quantities over a range is crucial.

For the function \( \sqrt{4-x^2} \), this represents a semicircle. The formula \( \frac{1}{2}\pi r^2 \) helps us find the area of this semicircle. Here, the radius \( r \) is \(2\), so the area comes to \(2\pi\). By subtracting the integral of the constant \(-2\), it represents adjusting this area for the whole interval \([-2, 2]\). The calculation of both elements gives you the integral's result, signified by \(2\pi - 8\), which represents the area under \( f(x) \). This teaches us not only about integrating simple curves but also about finding areas in more complex scenarios.