Question

5\. Suppose that \(f\) is continuous and that \(\int _ { 0 } ^ { 3 } f ( z ) d z = 3\) and \(\int _ { 0 } ^ { 4 } f ( z ) d z = 7\) Find each integral. (a) \(\int _ { 3 } ^ { 4 } f ( z ) d z \quad \quad\) (b) \(\int _ { 4 } ^ { 3 } f ( t ) d t \)

Short answer

\(\int _ { 3 } ^ { 4 } f ( z ) d z = 4\) and \(\int _ { 4 } ^ { 3 } f ( t ) d t = -4\)

Step-by-step solution

Use Additivity Property

The additivity property of integration says that the integral of \(f\) from \(a\) to \(c\) plus the integral of \(f\) from \(c\) to \(b\) equals the integral from \(a\) to \(b\). We can apply this to the given problem. We know that \(\int _ { 0 } ^ { 3 } f ( z ) d z = 3\) and \(\int _ { 0 } ^ { 4 } f ( z ) d z = 7\). From this, it follows that \(\int _ { 3 } ^ { 4 } f ( z ) d z = \int _ { 0 } ^ { 4 } f ( z ) d z - \int _ { 0 } ^ { 3 } f ( z ) d z\).

Execute Operation

Following from step 1: \(\int _ { 3 } ^ { 4 } f ( z ) d z = 7 - 3\). Now you can execute the operation to get the value of the integral.

Use Directionality Property

The property of directionality or change of variables in integrals states that the integral from \(a\) to \(b\) is the negative of the integral from \(b\) to \(a\). We use this property to find \(\int _ { 4 } ^ { 3 } f ( t ) d t\). This simplifies to \(-1 * \int _ { 3 } ^ { 4 } f ( t ) d t\). Take the result from step 2 to calculate this integral.

Key concepts

Additivity Property of Integration

An essential principle in calculus is the additivity property of integration. This property states that if you have a function that's continuous over an interval, you can break this interval into smaller parts, integrate the function over each part, and then sum these integrals to get the integral over the entire interval.

Here's how it works using the given exercise: We have two separate integrals of the same function, \(f\), over adjoining intervals, \(\int _ { 0 } ^ { 3 } f ( z ) d z = 3\) and \(\int _ { 0 } ^ { 4 } f ( z ) d z = 7\). To find the integral from 3 to 4, we take advantage of the additivity property. By subtracting the integral from 0 to 3 from the integral from 0 to 4, we isolate the integral from 3 to 4, obtaining \(\int _ { 3 } ^ { 4 } f ( z ) d z\).

This property, like a mathematical version of cutting and reassembling a string, lets us piece together information we have to find out what we don't know—an immensely helpful strategy when dealing with continuous functions.

Integral Operation Execution

Once you've applied the additivity property, the next step is execution of the integral operation. In our example, once the property is applied, we are left with a basic arithmetic operation: \(\int _ { 3 } ^ { 4 } f ( z ) d z = 7 - 3\). This simplifies to 4, which is the value of the integral. It shows that calculating an integral does not always involve complex calculus techniques; at times, it's reduced to simple subtraction.

Integral operation execution can also include employing various techniques like substitution or integration by parts when the integrale itself cannot be simplified by simple properties. Here, our problem doesn't require such steps, but it's critical to be equipped with these tools for more complicated integrals. Keeping operations straightforward and understanding when to use different techniques can turn a convoluted problem into a solvable one.

Directionality Property of Integrals

Directionality in integrals can be a source of confusion, but it's a powerful concept once understood. It deals with the orientation of integration limits. In calculus, if you reverse the limits of an integral, the sign of the integral changes. This is because integration is related to the accumulation of values over an interval, and switching the direction alters the perspective of accumulation.

The directionality property is in full display when we tackle part (b) of the given exercise: \(\int _ { 4 } ^ { 3 } f ( t ) d t\). Applying the property, we find that it equals \( -1 * \int _ { 3 } ^ { 4 } f ( t ) d t\). Now, remember the result of part (a), which found the integral from 3 to 4 to be 4. By plugging this value into our expression, we find \(\int _ { 4 } ^ { 3 } f ( t ) d t = -4\), reflecting the change in direction. Fully grasping this principle will prevent sign errors and lead to accurate calculation of integrals in various contexts.