Problem 13 - Thomas Calculus

Question

Reducing a double to a single integral By changing the order of integration, show that the following double integral can be reduced to a single integral: $$ \int_{0}^{x} \int_{0}^{u} e^{m(x-t)} f(t) d t d u=\int_{0}^{x}(x-t) e^{m(x-t)} f(t) d t $$ Similarly, it can be shown that $$ \int_{0}^{x} \int_{0}^{v} \int_{0}^{u} e^{m(x-t)} f(t) d u d v=\int_{0}^{x} \frac{(x-t)^{2}}{2} e^{m(x-t)} f(t) d t $$

Step-by-Step Solution

Answer

By changing the order of integration and integrating with respect to the new limits, the double and triple integrals are reduced to single integrals.

Step 1: Understand the given integral

The problem presents two integrals and tasks you with transforming them from a double and triple integral form to a single integral form. The first integral is a double integral: \[ \int_{0}^{x} \int_{0}^{u} e^{m(x-t)} f(t) \, dt \, du \] You are asked to show how this can be reduced to:\[ \int_{0}^{x}(x-t) e^{m(x-t)} f(t) \, dt \]

Step 2: Change the order of integration in the first problem

The limits of integration for the inner integral are from 0 to \( u \). We change the order of integration by considering the region of integration:1. In the original order, \( 0 \leq t \leq u \leq x \).2. In the reversed order, for each \( t \), \( t \leq u \leq x \).Thus, change the inner integral limits from \( u\) (for \( t\)) to \(x \) (for \( u \)), with the new limits for \( t \) ranging from 0 to \( x \).

Step 3: Solve new limits integral for the first problem

With the new limits, the integral becomes:\[ \int_{0}^{x} \int_{t}^{x} e^{m(x-t)} f(t) \, du \, dt \]Perform the integration with respect to \( u\):\[ = \int_{0}^{x} e^{m(x-t)} f(t) \cdot (x-t) \, dt \]This matches the given single integral form.

Step 4: Understand the triple integral in the second problem

The second problem gives a triple integral:\[ \int_{0}^{x} \int_{0}^{v} \int_{0}^{u} e^{m(x-t)} f(t) \, du \, dv \] We are asked to reduce this to:\[ \int_{0}^{x} \frac{(x-t)^{2}}{2} e^{m(x-t)} f(t) \, dt \]

Step 5: Change order of integration for the second problem

Start by considering the region of integration for the original triple integral:1. Original order: \( 0 \leq t \leq u \leq v \leq x \).2. Reversed order for integration: \( t \leq u \leq v \leq x \).The new integral arrangement is to perform integration with respect to \( u \), followed by \( v \), with limits adjusted accordingly.

Step 6: Solve new limits integral for the second problem

Once the order is changed, integrate with respect to \( u \) first, followed by \( v \). For the \( u \) integration:\[ \int_{t}^{x} (e^{m(x-t)} f(t)) \, du = (x-t) \, e^{m(x-t)} f(t) \]Then integrate over \(v \):\[ \int_{t}^{x} (x-t) \, dv = \frac{(x-t)^2}{2} \]Resulting in:\[ \int_{0}^{x} \frac{(x-t)^{2}}{2} e^{m(x-t)} f(t) \, dt \]

Key Concepts

Order of Integration

When evaluating a multiple integral, such as double or triple integrals, the order of integration refers to the sequence in which the integrations are performed over the different variables. Changing the order of integration can simplify the process and sometimes even result in a reduction from a multiple to a single integral. Consider a double integral:

• Original order: Integrate with respect to \( t \) first, then \( u \)
• Reversed order: Integrate with respect to \( u \) first, then \( t \)

This process involves understanding and possibly visualizing the region over which the function is being integrated, ensuring it aligns with the new order of integration.

Triple Integral

A triple integral involves integration over three variables, expanding upon double integrals by adding another layer of complexity. Like double integrals, the solution can often be simplified by changing the order of integration. Let us take a look at a typical triple integral, initially set up as:

• \( 0 \leq t \leq u \leq v \leq x \)

The reversed order turns into:

• \( t \leq u \leq v \leq x \)

Such rearrangement can help to reduce the computational workload or even transform the triple integral into an easier-to-handle form, like the single integral shown in the exercise.

Single Integral Reduction

Reducing a double or triple integral into a single integral is a powerful method for simplifying calculations. By cleverly altering the order of integration and understanding the corresponding integration limits, it's possible to express complicated multiple integrals as simpler single integrals. For instance, the exercise shows that a double integral can become:

• \( \int_{0}^{x} (x-t) e^{m(x-t)} f(t) \, dt \)

This simplified form requires fewer calculations and can often be solved directly, making it an advantageous method for tackling complex integrations.

Integration Limits

Integration limits define the boundary values for each variable in an integral. They are crucial in ensuring that the entire region of interest is covered during integration. When you change the order of integration, these limits need to be adjusted to reflect this new order correctly.

• For a double integral, swapping the limits of \( u \) and \( t \) changes how the variables interact.
• For a triple integral, the limits must also align with any change in the hierarchical integration order, such as moving from \( du \, dv \, dt \) to \( dt \, du \, dv \).

Correctly determining the integration limits after a change in order ensures that the integral remains over the same region and produces an accurate result.