Question

Evaluate the triple integrals. $$\int_{x=1}^{2} \int_{z=x}^{2 x} \int_{y=0}^{1 / z} z d y d z d x$$

Short answer

The result is 1.5

Step-by-step solution

Understand the Limits of Integration

Determine the order of integration and the boundaries for each variable. The integrals are nested as follows: the outer integral is \(\bigg( \int_{x=1}^{2} \bigg)\), the middle integral is \(\bigg( \int_{z=x}^{2x} \bigg)\), and the inner integral is \(\bigg( \int_{y=0}^{\frac{1}{z}} z \bigg)\).

Integrate with respect to y

Integrate the innermost function \(z\) with respect to y: \[ \int_{y=0}^{1/z} z dy = z \cdot y \bigg|_0^{1/z} = z \cdot \left(\frac{1}{z} - 0\right) = 1 \]

Integrate with respect to z

Substitute the result from the previous step into the middle integral and integrate with respect to z: \[ \int_{z=x}^{2x} 1 dz = z \bigg|_x^{2x} = 2x - x = x \]

Integrate with respect to x

Finally, perform the outermost integral with respect to x: \[ \int_{x=1}^{2} x dx = \frac{x^2}{2}\bigg|_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = 2 - 0.5 = 1.5 \]

Key concepts

limits of integration

When working with triple integrals, it's crucial to understand the limits of integration. These limits specify the range over which you integrate for each variable. For the given problem, the outermost limit is for x, ranging from 1 to 2. The middle limit pertains to z, ranging from x to 2x. The innermost limit is for y, ranging from 0 to \(\frac{1}{z}\).
This nesting of integrals ensures that each level of integration builds on the previous one. By clearly understanding these limits, you ensure precise and accurate integration.

iterated integration

Iterated integration is a method where you perform integration step by step from the innermost to the outermost integral. In this problem, we first integrate with respect to y, then z, and finally x.
This approach simplifies the complex problem into manageable parts.
Always start with the innermost integral and work your way outward, ensuring each step is completed correctly before moving to the next.

order of integration

The order of integration refers to the sequence in which you perform the integrations. For the given problem, the order is dy, dz, and then dx.
Changing the order of integration can sometimes simplify the problem or make it more complex. However, always ensure that the limits of integration match the specified order.

calculus

Calculus provides the tools needed to solve integrals, including derivatives and integrals. Triple integrals are an advanced application of these tools, allowing for the calculation of volumes and mass in three-dimensional space.
Understanding fundamental calculus concepts is essential. This will help you easily comprehend and solve more complex integrals.

evaluation of integrals

Evaluating integrals involves performing the integration process correctly and systematically. In this exercise, we followed these steps:

• Integrated z with respect to y.
• Substituted the result into the integral with respect to z and integrated.
• Finally, integrated the result with respect to x.

This systematic approach ensures that each integral is evaluated correctly and the solution obtained is accurate.