Surface Integrals of Scalar Valued Functions

Accumulating a scalar function over a surface in space — the mass of a curved sheet from its density — via a parameterization's area element.

Active Calculus Multivariable, Surface Integrals of Scalar Valued Functions

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3 exercises

Activity 13.12.2

In this activity, we will try to understand the scalar surface integral by looking at whether the value of the scalar surface integral will be positive, negative, or zero over common surfaces. In each part below, you are given a function and a surface. For each surface, first draw a plot of the surface and make sure you have labeled a proper scale for each coordinate direction. Then reason if the…

2.5

Activity 13.12.3

In this activity, we will be explaining each of the properties from Properties of Scalar Line Integrals in the context of a new analogy. We have just purchased a plot of land that spans two mountains, Sugar Mountain and Spice Mountain. We will label the plot of land on Sugar Mountain $S_1$ and the plot of land on Spice Mountain $S_2$ . Unfortunately there are two types of toxic organisms on the surface of…

2.5

Activity 13.12.4

Compute the value of the scalar surface integral in this part. That is, compute $\iint_{S_3} x+z \, dS$ where $S_3$ is the disc of radius one centered at $(1,0,0)$ on the plane $x=1$ . Explain why your answer makes sense geometrically.

2.5

Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.