Higher_Mathematics_Part_2

1

Conclusion. The second type of improper integral ∫ dx is convergent

0 xλ

if λ < 1 and is divergent if λ ≥ 1

Determine whether the improper integrals are convergent or divergent. Evaluate the convergent ones.

Determine whether the improper integrals are convergent or divergent

Topic 8. Application of the definite integral

Finding the area of a region. How to find the arc length. Computing volume by parallel cross sections. Computing the volume of a solid of revolution. The centroid of a plane region. Work.

Literature: [1, section 9], [2, section 2, ch. 2.2], [4, section 7, §24], [6, section 10], [7, section 12], [9, § 41].

8.1. Computing areas

8.1.1. Area in rectangular coordinates

Consider the area of the region bounded by y = f (x) ( f (x) ≥ 0 ) and the

x- axis from x = a to x = b, as shown in fig. 2.5. The area S of the entire region is found by

b

S = ∫ f (x)dx.

a

164

However, if a function changes the sign finitely in [a; b] (fig. 2.6) the area is found according to the following formula

the interval [a, b]. Let S be the region between the curve y = f1 (x) and the curve y = f 2 (x) for x in [a, b], as shown in fig. 2.7. The area of S is then given by

b

S = ∫[ f2 (x) − f1(x)]dx

a

It should be obvious to you that knowing the points of intersection is important in determining the limits of integration. For instance, the sketch of the region appears in fig. 2.8. To find where the curves intersect, we solve the

Sometimes area can more easily be determined by summing areas of horizontal elements rather than vertical elements. Let S be the region between

the curve x = g1 ( y) and the curve x = g 2 ( y) for y in [c, d], as shown in fig. 2.9 and 2.10. The area of S is then given by

d

S = ∫[g2 ( y) − g1 ( y)]dy.

c

165

The length l in polar coordinates of the curve ρ = ρ(ϕ) for ϕ in [ϕ1, ϕ2] is equal to

8.3. Volume of a solid

8.3.1. Computing volume by parallel cross sections

The volume V of a spatial region, a “solid”, can be expressed as a definite integral of cross-section area S(x),

b

V = ∫ S (x)dx

a

as shown in fig. 2.12. So, to find the volume of some solid, follow these steps:

1.Choose a line to serve as an x axis.

2.For each plane perpendicular to that axis find the area of the cross section of the solid made by plane. Call this area S(x).

3.Determine the limits of integration, a and b, for the region.

4.Evaluate the definite integral ∫b S(x)dx .

a

8.3.2. Computing the volume of a solid of revolution

Let y = f(x) be a continuous function such that f(x) ≥ 0 for x in the interval [a, b]. The curve y = f(x) is revolved around the x axis to form a solid of revolution (see fig. 2.13). The volume V of a solid is then given by

curves y = f1(x) and y = f2(x) and above [a, b]. The region S is revolved around the x axis to form a solid of revolution. The volume V of a solid is then given by

b

V = π∫[ f22 (x) − f12 (x)] dx.

a

8.4. Area of a surface of revolution

Let y = f(x) be a continuous function such that f(x) ≥ 0 for x in the interval [a, b] and f(x) has a continuous derivative. A formula for surface area P if y = f(x)