Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}(y+1) d A $$

Step-by-Step Solution

Answer

The solid is a triangular prism with its base as the rectangle \(0 \leq x \leq 2, 0 \leq y \leq 3\) and height \(y + 1\).

Step 1: Understanding the Region of Integration

The problem describes a rectangle defined by the limits \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). This means the base of the solid lies in the plane \((x, y)\) and is a rectangle with width 2 and height 3.

Step 2: Analyzing the Integrand

The integrand \(y + 1\) represents the height of the solid above each point \((x, y)\) in the region \(R\). As \(y\) varies from 0 to 3, the height changes from 1 to 4, indicating the solid is a prism with a slanted top surface (since it linearly increases with \(y\)).

Step 3: Visualizing the Solid

The solid can be visualized as a three-dimensional shape above the rectangle \(R\) on the \((x, y)\) plane. At \(y = 0\), the height is 1 across the entire width of the rectangle. As you move towards \(y = 3\), the height increases linearly up to 4, creating a slanted plane surface which forms the top of the prism.

Key Concepts

Volume of Solids

The concept of the volume of solids is crucial in understanding double integrals. Double integrals can be used to find the volume of a solid that lies above a region in the plane. In this exercise, the solid lies above a rectangle in the \(x, y\) plane, with limits \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). By integrating the function \(y + 1\) over this region, we determine the volume of the solid. This is because the integrand \(y + 1\) gives the height of the solid at each point within the base region. The integral essentially "adds up" infinitely many infinitesimally small volumes to obtain the total volume of the prism above the rectangle.

Region of Integration

The region of integration refers to the specific area over which the integration takes place. In this exercise, the region is a rectangle described on the \(x, y\) plane with boundaries \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\). This rectangular region serves as the base of the solid whose volume we calculate.

• The width of the rectangle is 2, as determined by the range of \(x\).
• The height is 3, given by the range of \(y\).

Understanding this base is essential because the integration is performed over this entire region, ensuring all points within it contribute to the final calculation of the solid's volume.

Prism Shape

A prism shape in this context refers to the type of solid above the region of integration. In simple terms, a prism is a three-dimensional shape where two opposite ends are identical, and the sides are parallelograms. Here, the base of the prism is our defined rectangle in the \(x, y\) plane. As the function \(y + 1\) determines the height for each \(x, y\), it results in a solid that has a linearly changing height when moving along the \(y\) axis. This gives the solid a particular look:

• The height is minimal at \(y = 0\) and becomes maximal at \(y = 3\).
• The top surface is not horizontal, due to the height variation, leading to a characteristic slanted top common in prismatic shapes.

Slanted Plane

A slanted plane refers to the inclined top surface of the prism, differentiating it from a flat horizontal plane. In geometric terms, it means the plane is not parallel to the base plane and inclines as \(y\) increases. The equation \(y + 1\) controls the rise in height across the rectangular base. This phenomenon is due to an important aspect:

• As \(y\) increases, the contribution to the height from \(y + 1\) also increases, creating a slant.
• At \(y = 0\), the height begins at 1, and by \(y = 3\), it reaches 4, showcasing a linear elevation.

Thus, the plane over the base is slanted, generating a simple yet visual way of understanding how the volume accumulates at varying degrees over the region.