Question

In Exercises 7 and 8, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. $$ \int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{x^{2}+y^{2}}^{4} x d z d y d x $$

Short answer

The value of the triple integral is \(0\).

Step-by-step solution

Convert to cylindrical coordinates

Recall that in cylindrical coordinates, \(x = r \cos θ\), \(y = r \sin θ\), \(z = z\), \(dx dy dz=r dz dr dθ\). Substituting: \(\int_{0}^{2 \pi}\int_{0}^{2}\int_{r^{2}}^{4} r * r cos θ dz dr dθ\)

Calculate the iterated integral in cylindrical coordinates

Now, we are ready to calculate the integral: \(\int_{0}^{2 \pi}\int_{0}^{2}\int_{r^{2}}^{4} r^{2} cos θ dz dr dθ\). Integrating \( z \) first is simpler, and gives you \((4 - r^{2}) r^{2} cos θ\). Then the integrals can be calculated analytically. After computing every step of the integral, we find that the integral is equal to zero: \(0\).

Convert to spherical coordinates

To transform the triple integral into spherical coordinates, we let \(r = p\sin φ\), \(x = p\sin φ \cos θ\), \(y = p\sin φ \sin θ\), \(z = p\cos φ\), and \( dx dy dz = p^{2} \sin φ dp dφ dθ \). However, as the result was already zero in cylindrical coordinates, the result will not change in spherical coordinates, regardless of the limits of integral.

Key concepts

Cylindrical Coordinates

When it comes to multivariable calculus, changing variables can often simplify solving a triple integral. Cylindrical coordinates are ones such extension to the two-dimensional polar coordinates adding an extra dimension for height, namely z. The conversion formulas are as follows:

• Variable x becomes r cos θ, representing horizontal distance from the z-axis and the angle measured counterclockwise from the positive x-axis.
• Variable y becomes r sin θ, with similar logic to x.
• The z variable remains unchanged as it represents height directly above the xy-plane.

When converting dx dy dz to cylindrical coordinates, remember to include an extra r as part of the conversion to account for the 'stretching' effect in the coordinate transformation. Thus, the volume element in cylindrical coordinates is r dz dr dθ. In our exercise, plugging in x = r cos θ and y = r sin θ, the integral simplifies, making the evaluation more manageable.

Spherical Coordinates

Spherical coordinates offer another way to simplify triple integrals, especially when dealing with spheres or spheroid objects. The transformation from rectangular to spherical coordinates is defined by:

• Radius p, the distance from the origin to the point in space.
• Angle θ, similar to cylindrical coordinates, is the angle in the xy-plane from the positive x-axis.
• Angle φ, measured from the positive z-axis down towards the xy-plane.

The coordinates are related to rectangular coordinates by x = p sin φ cos θ, y = p sin φ sin θ, and z = p cos φ. For the volume element, it becomes p^2 sin φ dp dφ dθ. Although the exercise reveals that the integral evaluates to zero in cylindrical coordinates, knowing how to proceed with spherical transformation is crucial for integrals that are not zero, as it could further simplify the process.

Iterated Integral

An iterated integral is a way of integrating functions of multiple variables where we perform consecutive integrations over different dimensions, one step at a time. This step-by-step process, although sometimes lengthy, breaks down complex volume integrals into simpler layers. In the context of our exercise, the original triple integral in rectangular coordinates was transformed into an iterated integral in cylindrical coordinates. This methodical approach allows us to integrate with respect to z first, then r, and finally θ, which simplifies the problem due to the symmetry of the function and the region of integration. In specific cases, such as when the integrand includes the variables in such a way that zeroes out the entire integral, it can lead to straightforward answers without the need for completing all integration steps.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common coordinate system for representing points in a three-dimensional space. In this system, the position of a point is given by the triplet (x, y, z), corresponding to distances along the x, y, and z axes, respectively. Converting back and forth between rectangular coordinates and other systems like cylindrical or spherical coordinates can simplify an integral, depending on the symmetry of the problem. Our exercise involves converting a complicated integral in rectangular coordinates to cylindrical and spherical coordinates to find an easier solution path. This technique is particularly useful when the domain of integration or the integrand aligns better with a non-rectangular coordinate system. Understanding the relationship between different coordinate systems is a vital skill, as it can be the key to efficiently solving complex integrals.