SUNY Geneseo Department of Mathematics
Friday, November 11
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
Based on “Scalar Line Integrals” in section 5.2 of the textbook.
Geometrically, a scalar line integral is the area of a curved sheet between the xy-plane (or space of input variables if there are more than 2) and the surface of a function.
Evaluate a scalar line integral as an integral, although slightly different in form from the abstract line integral.
To make a little geometric sense out of line integrals, consider the function f(x,y) = 9 - x2 - y2, integrated around a circle of radius 2 centered at the origin. Based on geometric considerations, what should this integral be?
The integral is the surface area of a cylinder of radius 2 and height 5, thus 20π:
Evaluate the integral and check that it has the expected value.
Start by finding a parametric form, r(t), for the circle. Then plug it and its derivative into the “f(r(t))…” formula for a line integral:
Another example of a scalar line integral.
Time permitting, we’ll start combining vector-valued functions and multivariable functions to talk about “vector fields.”