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Magazine Chapter 5: Problem 4
Suppose that \(\int_{4}^{8} f(x) d x=6 .\) Evaluate \(\int_{8}^{4} f(x) d x\).
Short Answer
Expert verified
The value of \(\int_{8}^{4} f(x) \ dx \) is \(-6\).
Step by step solution
01
Understand the properties of definite integrals
A definite integral from \(a\) to \(b\), \(\int_{a}^{b} f(x) \ dx\), gives the net area under a curve \(f(x)\) from \(x = a\) to \(x = b\). However, if the limits of integration are swapped, the integral changes sign. This can be mathematically represented as: \( \int_{a}^{b} f(x) \ dx = -\int_{b}^{a} f(x) \ dx \).
02
Apply the property of reversing limits
Given that \(\int_{4}^{8} f(x) \ dx = 6\), and knowing that changing the limits of integration changes the sign, we can write: \( \int_{8}^{4} f(x) \ dx = -\int_{4}^{8} f(x) \ dx \).
03
Substitute the known integral value
Substitute the given integral value into the equation from Step 2: \( \int_{8}^{4} f(x) \ dx = -6 \).
04
Conclude the solution
Since reversing the limits of the integral results in taking the negative of the original integral, the evaluation of \(\int_{8}^{4} f(x) \ dx \) is \(-6\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Properties of Integrals
When dealing with definite integrals, there are a few key properties that are very helpful to understand. One such property is the behavior that occurs when you reverse the order of the limits of integration in a definite integral. Let's start with a simple example. If you have the integral from point \(a\) to point \(b\), noted as \( \int_{a}^{b} f(x) \ dx \), this integral represents the net area under the curve of the function \(f(x)\) from \(x = a\) to \(x = b\). If you switch these points around, reversing the order to \( \int_{b}^{a} f(x) \ dx \), the integral's value will become the negative of the original value.
- The property can be expressed as: \( \int_{a}^{b} f(x) \ dx = -\int_{b}^{a} f(x) \ dx \).
- This sign change is crucial because it reflects the direction along the x-axis in which you are accumulating the area under the curve.
- Understanding this property can help simplify calculations and solve problems by knowing that any reversal of limits causes only a sign change.
Integral Limits
Integral limits are the boundary values in a definite integral that indicate the region over which the area under the curve is calculated. The two numbers in a definite integral, expressed as \( \int_{a}^{b} f(x) \ dx \), are the limits, with \(a\) as the lower limit and \(b\) as the upper limit. They define the segment of the curve \(f(x)\) under consideration.
- These limits play a critical role, as they determine the starting and ending points of the area calculation along the x-axis.
- Changing limits from \(a\) to \(b\) or from \(b\) to \(a\) changes the integral's sign, as explained with the properties of integrals.
- The order of integration limits defines the direction in which we integrate the function, significantly influencing the integral's result.
Area Under a Curve
One of the primary applications of definite integrals is in calculating the area under a curve. This is often visualized as the total area between the curve of function \(f(x)\) and the x-axis, evaluated over a specific interval from \(a\) to \(b\).
- The definite integral \( \int_{a}^{b} f(x) \ dx \) can be seen as the accumulation of infinitesimally small rectangles under the curve, adding up to give the total area.
- If the curve is above the x-axis, the area is positive, whereas if it lies below, the area is considered negative. This difference is critical as the area under the x-axis actually gets subtracted when calculating net area.
- In scenarios where curves cross the x-axis, the total area can be found by breaking the integral into sections where the function is above or below the axis.
