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Dated: 30-10-2024

Examples

Example

Evaluate the following from \(A(1, 2)\) to \(B(3, 5)\)

\[I = \int_c(3ydx + (3x + 2y)dy)\]

Solution

From the inspection, we notice that the integrant1 is in the following pattern

\[P dx + Q dy\]
\[\because P = \frac{\partial z}{\partial x} = 3y \therefore \frac{\partial P}{\partial y} = 3\]
\[\because Q = \frac{\partial z}{\partial y} = 3x + 2y \therefore \frac{\partial Q}{\partial x} = 3\]
\[\because \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}\]

This is an exact differential.2

\[\therefore I = \int_c dz\]

So now, we need to find what that \(z\) is.

\[P = \int 3y dx = 3xy + f(y)\]
\[Q = \int (3x + 2y) dy = 3xy + y^2 + g(x)\]

comparing \(P\) and \(Q\), we get

\[z = 3xy + y^2\]
\[\therefore I = \int_{(1, 2)}^{(3, 5)} d (3xy + y^2)\]
\[= \left[3xy + y^2\right]^{(3, 5)}_{(1, 2)}\]
\[= 60\]

Theorem

If \(P dx + Q dy\) is an exact differential2 then

  1. \(I = \int_c (Pdx + Qdy)\) is independent of the path of integration1
  2. \(I = \oint_c (Pdx + Qdy) = 0\)

Green's Theorem

Let \(P(x)\) and \(Q(y)\) be continuous functions3 within the region \(R\) including its boundary \(c\).
If its first partial derivatives4 are continuous then

\[\iint_R\left(\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}\right) dx dy = - \oint_C (P dx + Q dy)\]

Example

Evaluate the following at vertices \(O(0, 0)\), \(A(1, 0)\) and \(B(1, 2)\)

\[I = \oint (2x + y)dx + (3x - 2y)dy\]

Solution

By Previous Method

\(c_1\)
\[c_1 : y = 0 \therefore dy = 0\]
\[I_{c_1}=\int_{0}^{1}2xdx=\left[x^{2}\right]_{0}^{1}=1\]
\(c_2\)
\[c_2 : x = 1 \therefore dx = 0\]
\[I_{c_2}=\int_{0}^{2}(3-2y)dy=\left[3y-y^{2}\right]_{0}^{1}=2\]
\(c_3\)
\[c_3 : y = 2x \therefore dy = 2\]
\[I_{c_3}=\int_{1}^{0}\{4xdx+(3x-4x)2dx\}\]

therefore

\[I = I_{c_1} + I_{c_2} + I_{c_3} = 2\]

By Green's Theorem

\[I = \oint (2x + y)dx + (3x - 2y)dy\]
\[P = 2x + y \therefore \frac{\partial P}{\partial y} = 1\]
\[Q = 3x - 2y \therefore \frac{\partial Q}{\partial y} = 3\]
\[I=-\iint_{R}\left(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}\right)dxdy\]
\[I = 2 \iint_R dxdy = 2 A\]

Here \(A\) is the area of the triangle.

\[\because \text{Area} = 1\]
\[I = 2\]

References

Read more about notations and symbols.

  1. Read more about integrals. ↩↩

  2. Read more about exact differentials. ↩↩

  3. Read more about continuity. ↩

  4. Read more about partial derivatives. ↩