Skip to main content

Section 6 Wrap Up Questions

  1. Are the following statements true or false?

    1. If \(C \) is a curve and \(-C \) is the same curve oriented in the opposite direction, then for any scalar function \(f(x,y) \) we have \(\displaystyle \int_{-C} f(x,y)ds=\displaystyle -\int_{C} f(x,y)ds \text{.}\)
    2. If \(f(x,y)=k \) for some constant \(k \text{,}\) then \(\displaystyle \int_{C} f(x,y)ds=k|C| \) where \(|C| \) denotes the length of \(C \text{.}\)
    3. The integral \(\displaystyle \int_{C} f(x,y)ds \) is always positive since it measures the area of \(f(x,y) \) over the curve \(C \text{.}\)
    4. If \(C_1 \) is longer than \(C_2 \text{,}\) then \(\displaystyle \int_{C_1} f(x,y)ds>\displaystyle \int_{C_2} f(x,y)ds \text{.}\)
  2. Explain geometrically why if \(P \) and \(Q \) are two different points and \(C_1 \) and \(C_2 \) are curves between them, it is not necessary that \(\displaystyle \int_{C_1} f(x,y)ds=\displaystyle \int_{C_2} f(x,y)ds \text{.}\)
  3. If a curve \(C \) lies entirely on the contour \(f(x,y)=k \) of a function \(f(x,y) \text{,}\) what is the value of \(\int_{C} f(x,y)ds \text{?}\)
login