Chapter 6 Unit 5 定积分的几何应用定积分的几何应用

This section presents various geometric applications of the definite integral. We will show that area, volume and length of a curve can be represented as definite integrals. That is, “Area is the integral of the length of cross sections made by lines” and “ Volume is the integral of the areas of cross sections made by planes”.

1. Introduction a b x y o Method

See figure a b x y o If the quantity U satisfies the following conditions: (2) Method of element

element of U

Applications： The method of element has various applications, for instance, finding the area of a plane region, the volume of a solid, the length of a plane curve, the work done by a varying force that moves an object along a straight line. 2. Area of plane regions (1) Area of regions in rectangular coordinates

Example 1 Solution 1 See figure

Solution 2 See figure Example 2

SolutionSee figure

Example 3 Solution 1 See figure

Solution 2

Notice Although both approaches to finding the area of the region in example 3 are valid, the first one is more convenient. (2) Area of regions with parameter equations

Example 4 Solution The parametric equation of the ellipse is The part area of the first quadrant and multiply by 4, we obtain

(3) Area of regions in polar coordinates Since the polar coordinate is convenient for describing some curves, we introduce how to evaluate the areas by applying the polar coordinate. For example Solution See figure

The following examples apply this technique. Example 5

The total area = the area of the part in the first quadrant and multiply by 4, that is, Solution Example 6

By the symmetry, we have Solution

3. Computing volumes This section offers further practice in setting up a definite integral for the volume of a solid. We will find that “volume is the integral of cross-sectional area” (1) The volume of a solid of revolution solid of revolution The solid formed by revolving a region in a plane about a line in that plane is called a solid of revolution. See the following figures

circular cylinder circular conecircular truncated cone Question? Solution See figure

x y o The following examples we use the formula to find the volume of such a solid.

Example 7 Solution

Example 8 Solution

Similarly, The following examples we use the formula to find the volume of such a solid.

Example 9 Solution

Example 10 Solution See figure

(2) The volume of a solid with known area of cross sections made by planes Suppose that we wish to compute the volume of a solid S, and we know the area A(x) of each cross section made by planes in a fixed direction (see Fig.)

In order to evaluate the total volume we first approximate the volume of the region bordered by two parallel planes a distance dx apart

In other words, volume is the definite integral of cross-sectional area. Example 11 Solution See figure

Example 12 Solution See figure

4. The arc length of a plane curve Definition 1See figure

(1) Length of a curve in rectangular coordinates SolutionSee figure

Example 13 SolutionSee figure

Example 14 Solution

(2) Length of a curve with parametric equations

Solution

Example 15 Solution The length in the first quadrant

(3) Length of a curve in polar coordinates Solution

Example 16 Solution

Example 17 Solution See figure

This section presented various geometric applications of the definite integral. We showed that area, volume and length of the curve can be represented as definite integrals. Next section we will present various physical applications of the definite integral.