SUNY Geneseo Department of Mathematics

Introduction to Scalar Line Integrals

Friday, November 11

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Scalar Line Integrals

Based on “Scalar Line Integrals” in section 5.2 of the textbook.

Key Ideas

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.

Curved surface above curved line, with rectangles connecting line to its projection onto curved surface

Evaluate a scalar line integral as an integral, although slightly different in form from the abstract line integral.

Integral of F along curve is integral of F of R of T times magnitude of R prime

Examples

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:

Cylinder of radius 2 between surface 9 minus X squared minus Y squared and X Y plane has area 20 Pi

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:

Evaluating line integral around circle to find area of cylinder below 9 minus X squared minus Y squared

Next

Another example of a scalar line integral.

Time permitting, we’ll start combining vector-valued functions and multivariable functions to talk about “vector fields.”

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