Justify, with a written explanation or a mathematical reasoning and with a sketch of at least two different cases,
the following properties of integrals:
a) If f(x) is less than or equal to g(x) for a<=x<=b then
integrate from a to b of f(x)dx is less than or equal to integrate from a to b of g(x)dx
b)abs(integrate from a to b of f(x)dx) is less than or equal to integrate from a to b of abs(f(x))dx
a) To justify the property that if f(x) is less than or equal to g(x) for a <= x <= b, then the integral of f(x) from a to b is less than or equal to the integral of g(x) from a to b, we can use a combination of written explanation, mathematical reasoning, and a sketch of two different cases.
Let's start with the written explanation:
When f(x) is less than or equal to g(x) for every x in the interval [a, b], it means that the graph of f(x) is always below or at the same level as the graph of g(x) within the given interval. Hence, the area under the curve of f(x) from a to b is less than or equal to the area under the curve of g(x) from a to b.
Now, let's provide mathematical reasoning:
We can define a new function h(x) = g(x) - f(x). Since f(x) <= g(x) for every x in [a, b], this implies that h(x) >= 0 for every x in [a, b]. This means that h(x) represents the difference in height between the two functions, with h(x) being non-negative in the entire interval.
Now consider the integral of h(x) from a to b, denoted as ∫[a to b] h(x) dx. Since h(x) is non-negative in the interval, the integral represents the area between the curves of f(x) and g(x) from a to b. As h(x) >= 0, the integral of h(x) from a to b is also non-negative.
Using this information, we can write:
∫[a to b] h(x) dx >= 0
Expanding h(x) as g(x) - f(x):
∫[a to b] (g(x) - f(x)) dx >= 0
Rearranging the terms:
∫[a to b] g(x) dx - ∫[a to b] f(x) dx >= 0
Which can be rewritten as:
∫[a to b] f(x) dx <= ∫[a to b] g(x) dx
This shows that the integral of f(x) from a to b is less than or equal to the integral of g(x) from a to b.
To illustrate this property, we can provide a sketch of two different cases:
Case 1: When f(x) is always less than g(x) for every x in [a, b], the graph of f(x) lies entirely below the graph of g(x). In this case, the integral of f(x) from a to b is smaller than the integral of g(x) from a to b, as the area under the curve of f(x) is contained within the area under the curve of g(x).
Case 2: When f(x) is equal to g(x) for every x in [a, b], the two functions have the same graph within the interval. Consequently, the integral of f(x) from a to b will be equal to the integral of g(x) from a to b.
These examples demonstrate the validity of the property.
b) To justify the property that the absolute value of the integral of f(x) from a to b is less than or equal to the integral of the absolute value of f(x) from a to b, we can again provide a written explanation, mathematical reasoning, and a sketch of two different cases.
Let's start with the written explanation:
The absolute value of a function represents the distance of each point on the graph from the x-axis. By integrating the absolute value of f(x), we are calculating the total signed area between the graph of f(x) and the x-axis, considering both the area above and below the x-axis. On the other hand, by taking the absolute value of the integral of f(x), we disregard the sign of the integral and only consider the magnitude of the area.
Now, let's provide mathematical reasoning:
For any given function f(x), the integral of f(x) from a to b can have a positive, negative, or zero value. However, taking the absolute value of the integral ensures that we only consider the magnitude of the area, regardless of its sign.
To prove the property mathematically, we can consider two cases:
Case 1: When the integral of f(x) from a to b is non-negative:
In this case, the absolute value of the integral is equal to the integral itself. As a result, the property holds trivially:
|∫[a to b] f(x) dx| = ∫[a to b] f(x) dx <= ∫[a to b] |f(x)| dx
Case 2: When the integral of f(x) from a to b is negative:
In this scenario, the absolute value of the integral will negate the negative sign, resulting in a positive value. Therefore:
|∫[a to b] f(x) dx| = -∫[a to b] f(x) dx = ∫[a to b] |f(x)| dx
As the left-hand side (absolute value of the integral) and the right-hand side (integral of the absolute value) are equal, the property holds.
To illustrate this property, we can provide a sketch of two different cases:
Case 1: When the integral of f(x) from a to b is positive, the area between the graph of f(x) and the x-axis is entirely above the x-axis. The absolute value of the integral will give us the same value, representing the total area integrated. Similarly, the integral of the absolute value of f(x) will also give us the same value, as we consider the positive area only.
Case 2: When the integral of f(x) from a to b is negative, the area between the graph of f(x) and the x-axis is entirely below the x-axis. In this case, the absolute value of the integral will negate the negative sign, resulting in a positive value. The integral of the absolute value of f(x) will also give us the positive area.
These examples demonstrate the validity of the property.