Lecture ¹6

# Lecture ¹6. The double integral and its properties. The geometric meaning. The calculation of double integrals

Exercises for Lecture #6

The purpose of work:

The acquaintance with the double integral and its properties.

Theoretical information and examples

Content

2.1 The Double Integral in Rectangular Co-ordinates

2.1.1 Direct computation of double integrals

2.1.2 Setting up the limits of integration in a double integral

2.1 The Double Integral in Rectangular Co-ordinates

2.1.1 Direct computation of double integrals. The double integral of a continuous function f(x, y) over a bounded closed region S is the limit of the corresponding two-dimensional integral sum

where Dxi = xi+1 - xi, Dyk = yk+1 - yi and the sum is extended over those values of i and k for which the points (xi, yk) belong to S.

2.1.2 Setting up the limits of integration in a double integral. We make a distinction between two basic types of region of integration:

1) The region of integration S (Fig. 9) is bounded on the left and right hand sides by the straight lines x = x1 and x2 (x2>x1), from below and from above, by the continuous curves y = j1(x)(AB) and y = j2(x)(CD)[j2(x) ³ j1(x), each of which intersects the vertical line x = X (x1 £ X £ x2) at only one point (Fig. 9). In the region S, the variable x varies from x1 to x2 while the variable y (for constant x) varies from y1 = j1(x) to y2 = j2(x). Integral (1) may be computed by reduction to an iterated integral by the formula

where x is held constant when calculating

.

2) The region of integration S is bounded from below and from above by the straight lines y = y1 and y = y2 (y2 > y1) and from the left and the right hand side by the continuous curves x = y1(y) (AB) and x = y2(y) (CD) (y2(y) ³ y1(y), each of which intersects the parallel line y = Y (y1 £ Y £ y2) at only one point (Fig. 10)

As before, we have

where we consider f(x, y) in the last integral to be constant.

If the region of integration does not belong to any of the types, discussed above, then an attempt is made to break it up into parts, each of which does belong to one of these two types.

Example 1. Evaluate the integral

Solution.

Example 2. Determine the limits of integration of the integral

if the region of integration S (Fig. 11) is bounded by the hyperbola y² - x² = l and by the two straight lines x = 2 and x = -2 (we have in mind the region containing the origin).

Solution. The region of integration ABCD (Fig. 11) is bounded by the straight lines x=-2 and x=2 and by the two branches of the hyperbola

i.e., it belongs to the first type. We have:

Control question

1. Give the definition of the double integral of a continuous function f(x, y) over a bounded closed region S.