Problem 8 - Calculus Single Variable

Question

Suppose that \(\int_{3}^{5} f(x) d x=2\). Evaluate \(\int_{5}^{3}-4 f(x) d x\).

Step-by-Step Solution

Answer

The value of \(\int_{5}^{3} -4 f(x) dx\) is 8.

Step 1: Understanding the Property of Definite Integrals

Recall that if you reverse the limits of integration, the value of the integral changes sign. Thus, we have \(\int_{3}^{5} f(x) dx = 2\), which implies \(\int_{5}^{3} f(x) dx = -2\).

Step 2: Applying Scalar Multiplication to the Integral

For any constant \(c\), the integral \(\int_{a}^{b} c f(x) dx = c \int_{a}^{b} f(x) dx\). In this case, we have \(-4 \int_{5}^{3} f(x) dx\).

Step 3: Calculating the Final Integral

From Step 1, \(\int_{5}^{3} f(x) dx = -2\). Applying the result from Step 2, \(-4 \int_{5}^{3} f(x) dx = -4 \times (-2) = 8\).

Key Concepts

Properties of Integrals

Definite integrals are fascinating tools in calculus because they help determine the area under a curve between two points. One useful property of definite integrals is how their value changes when integration limits are swapped. If you have an integral from point \(a\) to \(b\), \(\int_{a}^{b} f(x) \, dx\), and you swap its limits to \(\int_{b}^{a} f(x) \, dx\), you simply change the sign of the integral's value.
For example, in our exercise, given that \(\int_{3}^{5} f(x) \, dx = 2\), by reversing the limits, we know \(\int_{5}^{3} f(x) \, dx = -2\). This property helps in simplifying integral calculations, especially when combining multiple integrals.

Integration Limits

Understanding integration limits is crucial for solving problems involving definite integrals. The limits in an integral, such as \(\int_{a}^{b} f(x) \, dx\), define the interval \([a, b]\) over which the function \(f(x)\) is evaluated. Swapping these limits not only changes the beginning and end points but also flips the sign of the computed integral result.
In many cases, recognizing how limits affect the integral can simplify complex calculations. In the example problem, the initial limits \([3, 5]\) were swapped to \([5, 3]\), changing the integral's value from 2 to -2. This simple reversal plays a significant role in solving various integration problems more efficiently.

Scalar Multiplication in Integration

Scalar multiplication is another important property in integral calculus that can simplify calculations. It states that a scalar (a constant number), when multiplied with an integral, can be factored out of the integral sign. In formal terms, for any constant \(c\), \(\int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx\). This property is particularly useful when dealing with integrals involving a product of constants and functions.
In our example, after reversing the limits and finding \(\int_{5}^{3} f(x) \, dx = -2\), multiplying by \(-4\) transforms it to \(-4 \times (-2) = 8\). This reflects how scalar multiplication allows for straightforward calculation of more complex integrals.