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# 数学代写|微积分代写Calculus代考|Volumes Using Cross-Sections

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## 数学代写|微积分代写Calculus代考|Volumes Using Cross-Sections

In this section we define volumes of solids by using the areas of their cross-sections. A cross-section of a solid $S$ is the planar region formed by intersecting $S$ with a plane (Figure 6.1). We present three different methods for obtaining the cross-sections appropriate to finding the volume of a particular solid: the method of slicing, the disk method, and the washer method.

Suppose that we want to find the volume of a solid $S$ like the one pictured in Figure 6.1. At each point $x$ in the interval $[a, b]$ we form a cross-section $S(x)$ by intersecting $S$ with a plane perpendicular to the $x$-axis through the point $x$, which gives a planar region whose area is $A(x)$. We will show that if $A$ is a continuous function of $x$, then the volume of the solid $S$ is the definite integral of $A(x)$. This method of computing volumes is known as the method of slicing.

Before showing how this method works, we need to extend the definition of a cylinder from the usual cylinders of classical geometry (which have circular, square, or other regular bases) to cylindrical solids that have more general bases. As shown in Figure 6.2, if the cylindrical solid has a base whose area is $A$ and its height is $h$, then the volume of the cylindrical solid is
$$\text { Volume }=\text { area } \times \text { height }=A \cdot h .$$
In the method of slicing, the base will be the cross-section of $S$ that has area $A(x)$, and the height will correspond to the width $\Delta x_k$ of subintervals formed by partitioning the interval $[a, b]$ into finitely many subintervals $\left[x_{k-1}, x_k\right]$.

## 数学代写|微积分代写Calculus代考|Slicing by Parallel Planes

We partition $[a, b]$ into subintervals of width (length) $\Delta x_k$ and slice the solid, as we would a loaf of bread, by planes perpendicular to the $x$-axis at the partition points $a=x_0<x_1<\cdots<x_n=b$. These planes slice $S$ into thin “slabs” (like thin slices of a loaf of bread). A typical slab is shown in Figure 6.3. We approximate the slab between the plane at $x_{k-1}$ and the plane at $x_k$ by a cylindrical solid with base area $A\left(x_k\right)$ and height $\Delta x_k=x_k-x_{k-1}$ (Figure 6.4). The volume $V_k$ of this cylindrical solid is $A\left(x_k\right) \cdot \Delta x_k$, which is approximately the same volume as that of the slab:
$$\text { Volume of the } k \text { th slab } \approx V_k=A\left(x_k\right) \Delta x_k \text {. }$$
The volume $V$ of the entire solid $S$ is therefore approximated by the sum of these cylindrical volumes,
$$V \approx \sum_{k=1}^n V_k=\sum_{k=1}^n A\left(x_k\right) \Delta x_k .$$
This is a Riemann sum for the function $A(x)$ on $[a, b]$. The approximation given by this Riemann sum converges to the definite integral of $A(x)$ as $n \rightarrow \infty$ :
$$\lim {n \rightarrow \infty} \sum{k=1}^n A\left(x_k\right) \Delta x_k=\int_a^b A(x) d x .$$
Therefore, we define this definite integral to be the volume of the solid $S$.

## 数学代写|微积分代写Calculus代考|Volumes Using Cross-Sections

$$\text { Volume }=\text { area } \times \text { height }=A \cdot h .$$

## 数学代写|微积分代写Calculus代考|Slicing by Parallel Planes

$$\text { Volume of the } k \text { th slab } \approx V_k=A\left(x_k\right) \Delta x_k \text {. }$$

$$V \approx \sum_{k=1}^n V_k=\sum_{k=1}^n A\left(x_k\right) \Delta x_k .$$

$$\lim {n \rightarrow \infty} \sum{k=1}^n A\left(x_k\right) \Delta x_k=\int_a^b A(x) d x .$$

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