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Section 3 Key Concepts

  • The line integral \(\int_{C} f(x,y)~ds \) of a scalar-valued function \(f(x,y) \) over a curve \(C \) is defined similarly to how the standard single variable definite integral is defined. Specifically:

    • Break up the curve \(C \) into small pieces and approximate each piece by a line segment.
    • For each line segment, construct a rectangle whose height is determined by the height of the graph of \(f(x,y) \) at some point on this segment and whose base is the line segment.
    • Construct the Riemann sum of the areas of these rectangles.
    • The integral \(\int_{C} f(x,y)~ds \) is defined to be the limit of this Riemann sum as the lengths of the line segments go to zero.
  • By the way it is defined, the line integral \(\int_{C} f(x,y)~ds \) of a scalar-valued function \(f(x,y) \) over a curve \(C \) gives the weighted area bounded between the curve \(C \) and the graph of the function \(f(x,y) \text{.}\)
  • If \(C \) is parameterized by \(\vec{r}(t) \) for \(a\leq t\leq b \text{,}\) then to calculate the \(\int_{C} f(x,y)~ds \) we use the formula
    \begin{equation*} \int_{C} f(x,y)~ds =\int_{a}^{b} f(\vec{r}(t)) ~ ||\vec{r}'(t)|| ~dt. \end{equation*}
    Note that if \(C \) is not already parameterized, this means you will need to find a parameterization for \(C \text{,}\) including the bounds.
  • Since \(\int_{C} f(x,y)~ds \) calculates weighted area, we can often approximate its value from a contour diagram of a specific graph.
  • Since geometrically a line integral of a scalar-valued function calculates weighted area, it does not depend on either the orientation of a curve or on the chosen parameterization. It does however depend upon the actual curve itself: that is, if \(C_1 \) and \(C_2 \) are two different curves between the same two points, we would expect the integrals over these curves to be different.
  • The concept of line integral of a scalar-valued function generalizes to functions of more variables. Such integrals also calculate area, though we cannot visualize it as easily since their graphs lie in higher dimensions.
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