4.4 Double Integrals over General Regions

Double Integrals over General Regions

Introduction

Double Integrals over General Regions

Horizontally and Vertically Simple Regions

A tetrahedron is a three-dimensional figure with four faces, each of which is a triangle. A picture of the tetrahedron $T$ with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ is shown in Figure. By using techniques from Section 1.7, we can find that an equation for this plane is

. However, we cannot directly use the methods we have developed so far involving double integrals to find the volume of a solid above the $xy$-plane and below a surface $z=f(x,y)$, as the base of the tetrahedron is not rectangular. This Preview Activity generalizes the work we have done so far with double integrals to understand how we can use a double integral to compute the volume of this tetrahedron. To do this, we will apply the Section 2.1.

As our first step in applying the Classic Calculus Approach, we approximate the tetrahedron by using three triangular prisms with bases parallel to the $yz$-plane. Specifically, the bases lie in the planes $x=0$, $x=1/3$, and $x=2/3$. We show these three triangular prisms in .

The volume of each triangular prism is the cross sectional area of the prism's base times the thickness of the prism. Find the cross sectional area for the three prisms shown in the figure.

On three separate plots, graph and label the regions $D$, $R$, and $B$.

For each double integral below, decide without calculation whether the double integral is positive, negative, or zero. Write a sentence or two to explain your answer for each part. -$\iint_D \; 1\; dA$-$\iint_D \; x \; dA$-$\iint_B \; x \; dA$-$\iint_R \; x \; dA$-$\iint_D \; x^2 \; dA$-$\iint_B \; 2-x^2 \; dA$-$\iint_R \; -e^x \; dA$-$\iint_B \; 1-e^y \; dA$-$\iint_D \; xy \; dA$

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

In , the region $D$ is broken into two regions, $V_1$ and $V_2$. Give inequalities for each of $V_1$ and $V_2$ that shows each region is vertically simple.

In , the region $D$ is broken into three regions $H_1$, $H_2$, and $H_3$. Give inequalities for each of $H_1$, $H_2$, and $H_3$ that shows that each region is horizontally simple.

For each region shown below, state whether the region is vertically simple, horizontally simple, both, or neither. State the appropriate inequalities to justify when a region is vertically simple or horizontally simple.

Work through this exercise and explain your reasoning step by step.