Question

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}\left(x^{2}+y^{2}\right) d A $$

Short answer

The solid is a paraboloid over the rectangle \( R \), with vertex at (0,0) and the highest point at (2,3,13).

Step-by-step solution

Identify the Area of Integration

The integration takes place over the rectangle \( R : 0 \leq x \leq 2, 0 \leq y \leq 3 \) in the \(xy\)-plane. This rectangle is bounded by the lines \(x=0\), \(x=2\), \(y=0\), and \(y=3\).

Understand the Function to Integrate

We need to integrate the function \( f(x, y) = x^2 + y^2 \). This function represents a paraboloid that points upwards with its vertex at the origin \((0, 0)\) in 3D space.

Visualize the Solid to Sketch

The solid we are concerned with has its base on the rectangle \( R \) in the \(xy\)-plane. Its height at any point \((x, y)\) is given by the function \(x^2 + y^2\), representing the surface of the paraboloid.

Sketch the 3D Solid

To sketch the solid, start by drawing the rectangle in the \(xy\)-plane. Then, imagine each point \((x, y)\) having a height of \(x^2 + y^2\). The resulting shape is a portion of a paraboloid, starting from \(z=0\) at the corners (since the smallest value of \(x^2 + y^2\) within the rectangle is at the origin) and increasing towards the edges.

View the Solid From Different Angles

When visualizing from different angles, the solid will appear to rise more steeply along the diagonal connecting \((0, 0)\) to \((2, 3)\) because \(x^2 + y^2\) is increasing. The highest point on the surface within this domain is at \((2, 3)\), where the height is \(2^2 + 3^2 = 13\). The cross-section parallel to the \(xy\)-plane is circular, with radii increasing as we move up.

Key concepts

Paraboloid

A paraboloid in mathematical terms can be understood as a surface in three-dimensional space. Imagine a shape similar to a rounded bowl or a satellite dish. The equation that describes a paraboloid is typically of the form \[ z = x^2 + y^2 \].This surface curves upwards, which means the values of \(z\), or height, increase as you move away from the origin in the \(xy\)-plane.

In the context of the given problem, the paraboloid has its vertex at the origin, or point \( (0,0,0) \). When we analyze the equation \[ f(x, y) = x^2 + y^2 \],we notice that every point \((x,y)\) will result in a non-negative value for \(z\) since squares can never be negative. Therefore, as you move away from the origin, the height of the paraboloid increases, creating that characteristic upward-facing bowl shape.

Volume of Solid

To find the volume of a solid using double integrals, imagine filling the space under a surface across a given region in the plane. The function we are given, \[ f(x, y) = x^2 + y^2, \]is integrated over a specified region to calculate the total volume beneath the paraboloid and above the rectangle in the \(xy\)-plane.

The double integral notation \[ \iint_{R}( x^2 + y^2 ) \, dA \] represents adding up all the tiny pieces of volume formed by the surface \( x^2 + y^2 \).Each "piece" can be visualized as having a small base area \(dA\) in the region \(R\) and a height given by \(x^2 + y^2\). Integrating across the entire region sums these small volumes to find the total volume under the paraboloid.

Area of Integration

The area of integration is where the calculations for volume take place. It's like the floor of the solid. In our example, this region \( R \) is a rectangle specified by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\).These boundaries mean the area of integration is comprised of all points that fall within \(x=0\) to \(x=2\) and \(y=0\) to \(y=3\) in the \(xy\)-plane.

In simple terms, if you were looking directly down on the solid from above, you would see a rectangular shape stretching from \( (0, 0) \) to \( (2, 3) \). This is the base over which the volume calculations will be performed. Understanding the area of integration is crucial for knowing where your calculations should start and end.

Sketching 3D Solids

Sketching a 3D solid, like the one formed by our paraboloid above the rectangle, starts with visualizing the base of the shape. First, draw the rectangle on the \(xy\)-plane, indicating where all calculations are focused.

Next, imagine a series of lines perpendicular to this base extending upwards, showing the heights marked by \(x^2 + y^2\). These heights form the surface of a paraboloid that rests above the rectangle. The result will look like a portion of a bowl with its surface slowly rising from each corner of the rectangle to its center.

• Start by drawing the rectangle given by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\) on a common plane.
• Then sketch how the height at each point \((x, y)\) is determined by \(x^2 + y^2\), beginning from \(z=0\) at each point in the base to a maximum height at point \((2, 3)\).
• The broader or further the point from the origin, the more the paraboloid rises.

Visualizing these elements from different angles helps in better understanding the solid's shape and how it forms under its specific conditions.