Lecture ¹7

# Lecture ¹7. The change of the variable in the double integrals. Practical application of double integrals

# Exercises for Lecture #7

The purpose of work:

The acquaintance with the change of the variable in the double integrals.

Theoretical information and examples

Content

2.2 Change of Variables in a Double Integral

2.2.1 Double Integral in polar co-ordinates

2.2.2. Double Integral in curvilinear co-ordinates

2.3 Computing Areas

2.3.1 Area in rectangular co-ordinates.

2.3.2 Area in polar co-ordinates.

2.2 Change of Variables in a Double Integral

2.2.1 Double Integral in polar co-ordinates. When passing in a double integral from rectangular co-ordinates (x, y) to polar co-ordinates (r, j), which are linked by

we have the formula

If the region of integration (S) is bounded by the half rays r = a and r = b (a < b) and the curves r=r1(j) and r=r2(j), where r=r1(j) and r=r2(j) [r1(j) £ r2(j)] are single-valued functions on the interval a £ j £ b, then the double integral may be evaluated using

where F(j, r) = f(r cos j, r sin j). In evaluating the last integral, we hold j constant.

If the region of integration does not belong to one of the kinds which have already been examined, it is broken up into parts, each of which is a region of a given type.

2.2.2. Double Integral in curvilinear co-ordinates. In the more general case, if it is required to pass in the double integral

from the variables x, y to the variables u, v, which are linked to x, y by the continuous and differentiable relations

which establish a (1,1) (and, in both directions, continuous) correspondence between the points of the region S of the xy-plane and the points of some region S' of the uv-plane, and if the Jacobian

retains a constant sign in the region S, then

The limits of the new integral are determined from general rules on the basis of the type of region S'.

Example 1. By passing to polar co-ordinates, evaluate

where the region S is a circle of radius R = 1 with centre at the origin (Fig. 12).

Solution. Setting x = r cosj, y = r sin j, we obtain

Since the co-ordinate r in the region S varies from 0 to 1 for any j, and j varies from 0 to 2p, it follows that

2.3 Computing Areas

2.3.1 Area in rectangular co-ordinates. The area of a plane region S is

If the region S is defined by the inequalities a £ x £ b, j(x) £ y £ y(x), then

2.3.2 Area in polar co-ordinates. If a region S in polar co-ordinates r and j is defined by the inequalities a £ j £ b, f(j) £ r £ F(j), then

2.4 Computing Volumes

The volume V of a cylinder bounded above by a continuous surface x = f(x, y), below by the plane z=0 and on the sides by a right cylindrical surface which cuts out of the xy-plane a region S (Fig. 94), is equal to

Control question

1. What formula is used when passing in a double integral from rectangular co-ordinates (x, y) to polar co-ordinates (r, j)?

2. What is the Jacobian?