Problem 20
Question
Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}\left(25-x^{2}-y^{2}\right) d A $$
Step-by-Step Solution
Verified Answer
The solid is a downward-opening paraboloid over a rectangular base.
1Step 1: Understand the Given Information
We're given a double integral that represents the volume under a specific surface. The surface is defined by the function \(z = 25 - x^2 - y^2\). Our task is to sketch the solid whose volume is represented by the double integral over the rectangle \(R = \{(x, y) : 0 \leq x \leq 2, 0 \leq y \leq 3\}\).
2Step 2: Identify the Function and the Region
The function given is \(z = 25 - x^2 - y^2\), which represents a paraboloid opening downward. The region \(R\) in the xy-plane is a rectangle bounded by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). This means the solid's base is this rectangular area in the xy-plane.
3Step 3: Determine the Limits of z
At any point \((x, y)\) in the rectangle \(R\), the value of \(z\) is determined by \(z = 25 - x^2 - y^2\). To understand the range of \(z\) over \(R\), calculate the values at the corners of \(R\): - At \((0,0)\), \(z = 25 - 0 - 0 = 25\).- At \((2,0)\), \(z = 25 - 4 - 0 = 21\).- At \((0,3)\), \(z = 25 - 0 - 9 = 16\).- At \((2,3)\), \(z = 25 - 4 - 9 = 12\).Thus, \(z\) ranges from 12 to 25.
4Step 4: Analyze the Solid's Surface
The surface \(z = 25 - x^2 - y^2\) over the region \(R\) forms the top boundary of the solid. The solid is therefore bounded above by this downward-opening paraboloid and below by the rectangle in the xy-plane.
5Step 5: Sketch the Solid
Begin by sketching the rectangle in the xy-plane which represents the base. Next, sketch the paraboloid \(z = 25 - x^2 - y^2\) with its maximum height of 25 at the origin, diminishing towards the edges of the rectangle. Make sure the paraboloid dips to z=12 at the farthest corner. This sketch will provide the visualization of the solid.
Key Concepts
ParaboloidVolume CalculationRectangular RegionSolid Sketching
Paraboloid
A paraboloid is a type of three-dimensional surface which can be thought of as being shaped like an open bowl or dome. In this problem, we encounter a specific type of paraboloid expressed by the equation \(z = 25 - x^2 - y^2\). This formula reveals that the paraboloid opens downward, meaning it curves towards the xy-plane. The surface reaches its highest point, or apex, at \((x, y, z) = (0, 0, 25)\), and declines as \(x\) and \(y\) increase.
As we move further away from the origin, the squares of \(x\) and \(y\) are subtracted from 25, causing \(z\) to decrease. This characteristic is crucial for identifying the boundaries of the solid formed by this surface, as we'll see in the following sections.
As we move further away from the origin, the squares of \(x\) and \(y\) are subtracted from 25, causing \(z\) to decrease. This characteristic is crucial for identifying the boundaries of the solid formed by this surface, as we'll see in the following sections.
Volume Calculation
Calculating the volume of a solid bounded by a surface and a region in the xy-plane involves determining a double integral. A double integral finds the accumulation of many infinitesimally small volumes over a region.
In our case, the region \(R\) is rectangular, making the integration straightforward. The given double integral \( \iint_{R}(25 - x^2 - y^2) \, dA \) calculates the volume enclosed by the paraboloid and the rectangle defined by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). This integral is evaluating the "height" or value of \(z\) over every infinitesimal area \(dA\) within \(R\). With the limits of integration matching the bounds of the rectangle, we can solve this integral to find the entire volume under the paraboloid and above the rectangular base.
In our case, the region \(R\) is rectangular, making the integration straightforward. The given double integral \( \iint_{R}(25 - x^2 - y^2) \, dA \) calculates the volume enclosed by the paraboloid and the rectangle defined by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). This integral is evaluating the "height" or value of \(z\) over every infinitesimal area \(dA\) within \(R\). With the limits of integration matching the bounds of the rectangle, we can solve this integral to find the entire volume under the paraboloid and above the rectangular base.
Rectangular Region
The rectangular region \(R = \{(x, y) : 0 \leq x \leq 2, 0 \leq y \leq 3\}\) is the base over which the paraboloid extends. This area in the xy-plane is easy to visualize as a flat rectangle. It defines the area within which our paraboloid creates a solid.
It's crucial to understand this base, as it determines the extent of the overhanging paraboloid. For the double integral, it establishes the limits for \(x\) and \(y\), guiding us in computing the volume and visualizing where the paraboloid meets the plane. By integrating over this region, we consider every possible point beneath \(z = 25 - x^2 - y^2\) within the rectangle.
It's crucial to understand this base, as it determines the extent of the overhanging paraboloid. For the double integral, it establishes the limits for \(x\) and \(y\), guiding us in computing the volume and visualizing where the paraboloid meets the plane. By integrating over this region, we consider every possible point beneath \(z = 25 - x^2 - y^2\) within the rectangle.
Solid Sketching
Sketching the solid involves combining the top boundary (the paraboloid) and the base (the rectangle). First, visualize the rectangular region \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). This is the flat area where the paraboloid rests.
Next, picture the paraboloid \(z = 25 - x^2 - y^2\). Begin at its apex at \((0,0,25)\) and extend it downwards over the entire rectangular base. The paraboloid slopes downwards, reaching its lowest point of \(z = 12\) at \((2,3)\). The goal is to understand this interaction spatially, where the paraboloid forms the upper surface settling onto the rectangular xy-plane base, encapsulating the volume we're interested in calculating. This conceptual sketching is pivotal in providing intuition behind how these geometric shapes form the complete solid.
Next, picture the paraboloid \(z = 25 - x^2 - y^2\). Begin at its apex at \((0,0,25)\) and extend it downwards over the entire rectangular base. The paraboloid slopes downwards, reaching its lowest point of \(z = 12\) at \((2,3)\). The goal is to understand this interaction spatially, where the paraboloid forms the upper surface settling onto the rectangular xy-plane base, encapsulating the volume we're interested in calculating. This conceptual sketching is pivotal in providing intuition behind how these geometric shapes form the complete solid.
Other exercises in this chapter
Problem 20
Evaluate by using polar coordinates. Sketch the region of integration first. \(\iint_{S} \sqrt{4-x^{2}-y^{2}} d A\), where \(S\) is the first quadrant sector of
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View solution Problem 21
Use spherical coordinates to find the indicated quantity. Volume of the smaller wedge cut from the unit sphere by two planes that meet at a diameter at an angle
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Find the Jacobian for the transformation from rectangular coordinates to spherical coordinates.
View solution