Question

Determine which value best approximates the volume of the solid between the \(x y\) -plane and the function over the region. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(f(x, y)=x y+2 ; R:\) quarter circle: \(x^{2}+y^{2}=9, x \geq 0, y \geq 0\) (a) 25 (b) 8 (c) 100 (d) 50 (e) -30

Short answer

The volume that best approximates the solid between the \(xy\)-plane and the function over the region is (a) 25.

Step-by-step solution

Interpret the Task

The goal is to find the volume of the solid that lies between the \(xy\)-plane and the function \(f(x, y) = xy + 2\) on the region R, which is a quarter circle with radius 3 in the first quadrant.

Sketch the Plane and the Region R

Draw the plane \(f(x, y) = xy + 2\). The gradient of the plane shows that it increases linearly as either \(x\) or \(y\) increases. Then, sketch the region R in the \(xy\)-plane, which is a quarter circle of radius 3 centered at the origin, \(x^2 + y^2 = 9\), with \(x, y \geq 0\).

Estimate the Volume

On a 3D plot, most of the volume of the solid under the function above the quarter circle is below the surface \(z = 9\). The quarter disk has an area of \(\frac{1}{4}\pi r^2 = \frac{9\pi}{4}\), which is less than 9. So the volume of the solid is less than \(9 \times 9 = 81\). Remember the constants (a), (b), (c), (d), and (e) are the approximate volume. Because the solid is always above the \(xy\)-plane, it is not a negative volume, eliminating choice (e). With an estimated upper bound of 81 for the volume, only choices (a) and (b), 25 and 8, are reasonable. Although we're just making a rough estimate, we know that most of the solid is above the height 2, because \(f(x, y) = xy + 2\). It seems reasonable that the volume is closer to 25 than to 8, and we choose (a) 25.

Key concepts

Double Integration

Double integration is a mathematical method used extensively to calculate the volume under a surface in three-dimensional space. Imagine taking a complex shape and slicing it into infinitely thin cross-sections. By adding up the areas of these cross-sections, we can approximate the volume of the shape. If the cross-sections are very small, the sum becomes an integral.

For a function like \( f(x, y) = xy + 2 \), you would integrate twice: first with respect to y and then with respect to x, or vice versa. As this function is continuous and the region of integration is simple (a quarter circle), we can transform the xy-coordinates into polar coordinates for easier computation. This transformation simplifies the double integral into a single iterated integral, where the function is integrated over a radius (r) and an angle (\( \theta \)).

Solid of Revolution

A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional shape around an axis. The resulting shape is symmetrical and can be computed mathematically using methods like the disk, washer, or shell method. However, in our context, the solid is not created by revolution but by the region under a function above a quarter circle.

The volume of a solid of revolution typically involves integration. For example, if we were to revolve the quarter circle around the x or y axis, we would typically use the disk method to find the volume, integrating \( \pi y^2 \) over the bounds of x, or vice versa.

Volume Estimation

Volume estimation is crucial when exact calculations are complex or unnecessary. For the given function \( f(x, y) = xy + 2 \), a precise numerical volume calculation involves setting up and evaluating a double integral, which can be substantial work. Instead, if we seek an approximation, we can estimate by looking at the behavior of the function and the size of the region.

In our exercise, the volume estimation involved observing that the solid lies below \( z = 9 \) and above the xy-plane. Since the volume is always positive and less than \( 9 \times 9 = 81 \), it guided us to the right answer, dismissing obviously incorrect choices. Estimation techniques might involve looking at bounds, using geometric shapes whose volume we can readily calculate (like cubes or cylinders), or considering the average height of the solid above the region.

3D Plotting

3D plotting is an invaluable tool in visualizing mathematical concepts and figures, especially when dealing with multiple-variable functions or complex solids. Modern computer software allows for the visualization of surfaces and solids that are difficult to imagine. For students and mathematicians alike, plotting provides insight into how functions behave in space and how solids take shape.

In this case, a 3D plot of the function \( f(x, y) = xy + 2 \) over the given region would reveal the slope of the plane it creates, and how it intersects with the xy-plane. This visual can clarify our volume estimation by anchoring our understanding of the shape and size of the solid defined by the specified region. Good 3D plotting skills enable learners to transition from abstract problem statements to tangible visual comprehension of mathematical objects.