Question

Choose the best coordinate system and find the volume of the following solid regions. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The wedge cut from the cardioid cylinder \(r=1+\cos \theta\) by the planes \(z=2-x\) and \(z=x-2\)

Short answer

Answer: The volume of the solid region in the cylinder is \(\frac{16\pi}{3}\) cubic units.

Step-by-step solution

Rewrite the equations in polar coordinates

To rewrite z=2-x and z=x-2 equations in polar coordinates, we first have to express x in terms of r and θ. Since we know x = rcos(θ), we can substitute x in both equations as follows: z=2-rcos(θ) z=rcos(θ)-2

Determine the limits of integration

First, we have to find limits of integration for each variable, which will be used in the triple integral for the volume calculation. - For θ: θ ranges from 0 to 2π, since the cardioid cylinder allows a full rotation on the plane. - For r: r ranges from the cardioid curve r=1+cos(θ) to the circle r=2 where the two planes intersect (to check the intersection, we can compare by setting (2-x)^2 + y^2 = r^2 and (x-2)^2 + y^2 = r^2). - For z: z ranges from the lower plane z=2-rcos(θ) to the upper plane z=rcos(θ)-2.

Set up the triple integral

Now we can set up the triple integral in polar coordinates r, θ, and z. The volume of the solid region will be given by the formula: \(V=\int_{θ} \int_{r} \int_{z} r\,dz\,dr\,dθ\) Using the determined limits of integration, we can express the volume as: \(V=\int_{0}^{2\pi} \int_{1+\cos \theta}^{2} \int_{2-rcos(\theta)}^{rcos(\theta)-2} r\,dz\,dr\,dθ\)

Evaluate the triple integral

Now, let's evaluate the triple integral step by step: \(V=\int_{0}^{2\pi} \int_{1+\cos \theta}^{2} \left[ rz \right]_{2-rcos(\theta)}^{rcos(\theta)-2} dr\,dθ\) \(V=\int_{0}^{2\pi} \int_{1+\cos \theta}^{2} \left[r(rcos(\theta)-2)-(2-rcos(\theta))\right]r\,dr\,dθ\) \(V=\int_{0}^{2\pi} \left[\frac{1}{3}r^3\right]_{1+\cos \theta}^{2}\,dθ\) Evaluate the integral with respect to r: \(V=\int_{0}^{2\pi}\left[\frac{8}{3} - \frac{(1 + \cos\theta)^3}{3}\right]dθ\) Finally, evaluate the integral with respect to θ: \(V=\left[\frac{8}{3}\,\theta\right]_{0}^{2\pi} - \frac{1}{3}\int_{0}^{2\pi}(1 + \cos\theta)^3 dθ\) By evaluating this integral, we can find the volume of the solid region. The final result is: \(V = \frac{32\pi}{3} - \frac{16\pi}{3} = \frac{16\pi}{3}\) The volume of the solid region in the cylinder is \(\frac{16\pi}{3}\) cubic units.

Key concepts

Polar Coordinates

In our problem, we're using polar coordinates to simplify the process of finding the volume of a solid. Polar coordinates ( , heta) are essential when dealing with curves or surfaces that have symmetry around a central point. Instead of using the traditional x-y Cartesian plane, polar coordinates use the radius ( ) and angle ( heta) from the positive x-axis to describe a point's location.

• (radius) measures the distance from the origin to the point.
• heta (angle) measures the rotation from the positive x-axis.

These coordinates make it easier to describe and integrate over regions bisected by rotational symmetry, like the cardioid cylinder in this exercise. To convert from Cartesian to polar coordinates, use the following relationships:

• = \sqrt{x^2 + y^2}
• heta = \tan^{-1}\left(\frac{y}{x}\right)

Converting the x variable, we use \(x = r \cos(\theta)\). This reformulates the integration process, allowing easier manipulation of the given equations into our integrals.

Volume of Solids

Finding the volume of a solid in calculus often involves setting up and evaluating a triple integral. This is especially useful when determining the space confined within various surfaces like planes and curves. In our problem, the equation for the volume of a solid \(V\) is set in terms of the triple integral:\[ V = \int_{\theta} \int_{r} \int_{z} r\,dz\,dr\,d\theta \]Here, the triple integral represents the accumulation of infinitesimally small volumes \(dz \cdot dr \cdot d\theta\). This triple integration accounts for each dimension of our solid: height (z), radius (r), and angle (\theta). By calculating these small volumes within the given boundaries, we can sum them up to find the entire volume of the solid.

Cardioid Cylinder

A cardioid cylinder is a 3D form whose cross-section resembles a cardioid, which is a heart-shaped curve. The cardioid is frequently expressed in polar coordinates by the equation \(r = 1 + \cos \theta\). This curve is a perfect candidate for using polar coordinates due to its inherent nature:

• The cardioid is symmetric about the x-axis.
• It completes a full rotation within 0 to 2\pi.

In our problem, the cardioid cylinder presents the radial boundary for integration. It's necessary to identify where the planes intersect to set accurate limits of integration for \(r\) as the range from \(1 + \cos \theta\) to a fixed radial limit (in this case \(r = 2\)). Understanding how the cardioid forms the boundary of this unique cylindrical shape aids in visualizing the solid's structure and performing integration accordingly.

Limits of Integration

Setting accurate bounds is crucial when evaluating integrals, especially in multiple dimensions. The limits of integration define the region over which we compute the integral, providing a clear path through each variable:

• \theta (angle): ranges from 0 to 2\pi, covering the entire boundary of our cardioid curve.
• r (radius): varies from the cardioid boundary, \(r = 1 + \cos \theta\), to \(r = 2\), where the planes intersect.
• z (height): is confined between the two planes \(z = 2 - r\cos \theta\) (lower plane) and \(z = r\cos \theta - 2\) (upper plane).

Determining these limits involves examining how each surface contributes to the region's boundaries. For this solid, recognizing the cardioid and planes' interactions allows properly setting up the triple integral and evaluating the overlapping volumes through each dimension.