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Magazine Chapter 6: Problem 31
In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s. $$ \int_{-1}^{2}\left(x^{2}-1\right) d x $$
Short Answer
Expert verified
The integral represents the net area, considering negative and positive areas on the intervals \([-1, 1]\) and \([1, 2]\).
Step by step solution
01
Understanding the Integral
The integral \( \int_{-1}^{2}(x^2 - 1) \, dx \) represents the net area between the curve \( y = x^2 - 1 \) and the x-axis, over the interval \([-1, 2]\). We aim to interpret this integral in terms of geometric areas without computing it.
02
Graph the Function
Plot the function \( y = x^2 - 1 \). This is a parabola that opens upwards, shifted downwards by 1 unit. Identify its intersections with the x-axis, which occur when \( x^2 - 1 = 0 \), so \( x = \pm 1 \). Thus, the curve intersects the x-axis at \( x = -1 \) and \( x = 1 \).
03
Identify Area Above and Below X-axis
The curve \( y = x^2 - 1 \) lies below the x-axis on the interval \([-1, 1]\) because the parabola dips below the x-axis in this region. It lies above the x-axis on the interval \([1, 2]\). Calculate the approximate shapes formed by the curve and consider these areas separately.
04
Analyze Each Area
On the interval \([-1, 1]\), the curve below the x-axis forms a symmetric area around the y-axis. From \( x = 1 \) to \( x = 2 \), the curve is above the x-axis. Calculate these separately: the area between \( -1 \) and \( 1 \) is below the curve, and the area between \( 1 \) and \( 2 \) is above it.
05
Determine Net Area
The area below the x-axis on \([-1, 1]\) will be negative when considering the definite integral, and the area above the x-axis on \([1, 2]\) will be positive. The definite integral value represents the net area, which is the sum of these two areas: \(-\text{(area below x-axis)} + \text{(area above x-axis)}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Area Under a Curve
The concept of area under a curve is fundamental to understanding definite integrals. When we consider the definite integral \[\int_{-1}^{2} (x^2 - 1) \, dx\] it represents the total area between the function graph of \[y = x^2 - 1\] and the x-axis over the interval from \(-1\) to \(2\). This area can be composed of parts that are above or below the x-axis and may include negative values when below.
- From \(-1\)to \(1\), the curve dips below the x-axis, creating a negative area.
- From \(1\)to \(2\), the curve rises above the x-axis, creating a positive area.
Interpretation of Integrals
Integrals, particularly definite integrals, have a profound significance in mathematics. They help us ascertain areas under curves and much more. The definite integral \[\int_{-1}^{2}(x^2 - 1) \, dx\] gives rise to a geometric interpretation.
- The area calculation is "net" because portions below the x-axis are negative, affecting the total sum.
- Thus, when areas below are subtracted from those above, it provides a combined measure of how much of the curve is on either side of the x-axis.
Graphical Representation
Visualizing integrals sheds light on their intuitive interpretation. For this particular integral, we graph the function \(y = x^2 - 1\). The resulting graph is a parabola. Crucially, it shows:
- Intersection points at \(x = -1 \) and \(x = 1\), where the graph touches the x-axis, marking transitions from positive to negative area.
- The curve dips under the x-axis initially, then rises above from \(x = 1\) to \(x = 2\).
Calculus Concepts
In the realm of calculus, integrals emerge as tools for measuring and understanding change and accumulation. At the heart of \[\int_{-1}^{2} (x^2 - 1) \, dx\] is the concept of finding net area. This ties into calculus principles such as:
- Fundamental Theorem of Calculus: Bridges the derivative and the integral by asserting that integration can "undo" differentiation.
- Limits: Explain the precision of calculating areas as the sum of infinite approaches."
- Differential Equations: Use integrals to express the accumulation of quantities that change over time.
