Problem 39 - Algebra 2

Question

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ h(x)=\sqrt{x^{2}} $$

Step-by-Step Solution

Answer

The total area under the curve \( h(x) = \sqrt{x^{2}} \) between -1 and 2 is 3 square units.

Step 1: Rewrite the Function

Notice first that \( \sqrt{x^{2}} \) can be simplified to |x|. So the function \( h(x) \) can be rewritten as \( h(x) = |x| \).

Step 2: Split the Interval

The function \( h(x) = |x| \) is defined differently for values of x that are less than 0 and values that are greater than or equal to 0. Thus, the integral should be split into two parts at x = 0: one from -1 to 0, and the other from 0 to 2.

Step 3: Integrate the Function

For the first part, since \( x \leq 0 \), \( |x|= -x \). Integrate this from -1 to 0. For the second part, since \( x \geq 0 \), \( |x|= x \). Integrate this from 0 to 2.

Step 4: Complete the Calculation

After calculating the two integrals, add the results together to get the total area under the curve between -1 and 2. Remember that when working with definite integrals, after calculating the antiderivative, you need to evaluate it at the upper limit and subtract the evaluation at the lower limit. After doing this for both intervals, add the results to get the area under the curve.

Key Concepts

Definite Integrals

Definite integrals are a key concept in integral calculus. They help us calculate the exact area under a curve over a given interval. While the indefinite integral gives us a family of functions, the definite integral results in a specific numerical value.

To find the definite integral, we need two main elements:

• The function to integrate, which is the mathematical equation representing the curve.
• The limits of integration, which specify the interval over which we calculate the area.

To compute a definite integral, you follow these steps:

1. **Find the Antiderivative**: The first step is to determine the antiderivative (also called the integral) of the function. This means finding a function whose derivative is the given function.
2. **Apply the Limits of Integration**: Once you have the antiderivative, evaluate it at the upper limit of the integral and at the lower limit. Subtract the value at the lower limit from the value at the upper limit.
3. **Calculate the Area**: The result of this subtraction gives you the area under the curve on the specified interval.

For the given exercise, the area between \(-1\) and \(2\) is calculated by splitting the interval into parts because of the absolute value function. More on this next!

Absolute Value Function

The absolute value function is a common piece in calculus and is unique because it behaves differently based on the input value.

The function is represented as \(|x|\), which means:

• If \(x\) is less than zero, \(|x| = -x\).
• If \(x\) is greater than or equal to zero, \(|x| = x\).

To solve integrals involving absolute values, you should separate the function into distinct segments where each segment is expressed as a linear function. This is necessary because applying the definition directly affects calculations over certain intervals.

In our exercise, \(h(x) = \sqrt{x^2}\) simplifies to \(|x|\). To calculate the integral from \(-1\) to \(2\), we split the integral based on \(x = 0\). So, from \(-1\) to \(0\), the function becomes \(-x\), and from \(0\) to \(2\), it turns into \(x\). This step ensures the area interpretation of the curve remains correct.

Understanding how \(|x|\) transforms can deeply enhance our calculus comprehension!

Area Under a Curve

Finding the area under a curve is a practical application of definite integrals in mathematics. It allows us to understand not just the behavior of a function but also to quantify regions in terms of their exact dimensions.

The area can be thought of as the sum of infinitesimally small rectangles under the curve. The definite integral adds up these rectangles across the interval to give a total area. This idea is central in various fields such as physics and economics where understanding relationships and quantitative measures is crucial.

To find the area under the curve for our exercise, we use the definite integral of \(|x|\) from \(-1\) to \(2\). However, as highlighted, due to the nature of absolute values, we must calculate the area in two parts. Each part corresponds to a section of \(x\) where the function form changes from \(-x\) to \(x\).

By computing the integrals separately for each part and summing them up, we assure that every portion of the function contributes correctly to the total area. This approach not only displays the area but clarifies the continuous nature of curves and integral calculus methods.