Problem 35 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) ASSORTED $$y=-\left(x^{2}+z^{2}\right)$$

Step-by-Step Solution

Answer

Sketch a downward-opening paraboloid with circular cross-sections in the \(xz\)-plane.

Step 1: Identify the Surface Equation

The surface given is described by the equation \(y = -(x^2 + z^2)\). This equation represents a paraboloid, which opens downward along the \(y\)-axis.

Step 2: Understand the Surface

The given equation \(y = -(x^2 + z^2)\) is in the form of a paraboloid. The minus sign indicates that as \(x\) and \(z\) increase in a positive or negative direction, \(y\) becomes more negative, forming a downward opening shape along the \(y\)-axis.

Step 3: Determine the Cross-Sections

For the plane \(y = c\), where \(c\) is a constant, the cross-sections parallel to the \(xz\)-plane are circles described by \(x^2 + z^2 = -c\). Each \(c < 0\) gives a real circle, decreasing in radius as \(c\) becomes more negative.

Step 4: Examine the Horizontal Cross-Section

When \(y = 0\), \(x^2 + z^2 = 0\). This means the only point on this cross-section is the origin \((0, 0, 0)\). As \(y\) decreases, the circular cross-section increases in size.

Step 5: Sketch the Paraboloid

Draw the \(y\)-axis vertically and \(x\)- and \(z\)-axes horizontally. Sketch a series of circles perpendicular to the \(y\)-axis that increase in size as \(y\) decreases, illustrating the downward-opening paraboloid.

Key Concepts

Surface Equation

The equation for a surface provides a foundation for understanding its shape and behavior in three-dimensional space. Here, we have the equation of a paraboloid: \[ y = -(x^2 + z^2) \] - The equation includes a negative sign which signifies that the paraboloid opens downward along the y-axis. - As x and z increase in any direction (positive or negative), their squares become larger, making y more negative. This particular type of paraboloid is sometimes called a concave paraboloid due to its downward opening nature. Typically, its central axis, in this case the y-axis, is the direction in which the surface extends outward. Such understanding becomes crucial for visualizing the entire surface and predicting how it intersects with different planes.

Cross-Sections

Cross-sections provide simplified slices of three-dimensional surfaces by fixing one variable. For the given paraboloid equation: \[ y = -(x^2 + z^2) \] we focus on cross-sections parallel to certain planes by setting y to a constant value c. - **Vertical Cross-Sections:** When a plane is parallel to the xz-plane (y = c), we get: \[ x^2 + z^2 = -c \] This represents circles in the xz plane, but only for c less than 0 can these be realized in real terms. - As c becomes more negative, the circles grow wider, illustrating a bigger radius of the cross-section.- **Horizontal Cross-Section:** When y equals zero, we only get a single point \[ x^2 + z^2 = 0 \] which is the origin point (0, 0, 0). As we pick y values that are negative, the size of the circular cross-sections increases, painting the picture of a continuously expanding circle as we move downward along the negative y-axis.

Surface Sketching

Sketching a paraboloid requires visualizing the shape outlined by its equation. Begin by identifying the main axes: y for vertical and x and z for horizontal directions. - **Drawing the y-axis:** The y-axis runs vertically. In our paraboloid, each cross-section grows as y decreases (becomes more negative). This axis is crucial for showing the depth of the paraboloid. - **Setting the x and z-axes:** These should run horizontally and are used to draw cross-section circles accurately. - **Illustrating cross-section growth:** Consider drawing several circles perpendicular to the y-axis. Start small near y = 0 and increase their radius with more negative y values to capture the downward opening. Remember, practice makes perfect when sketching 3D shapes. Focus on the increasing size and negative movement along the y-axis to depict the paraboloid opening correctly.