Area with Green's theorem practice problems
Green's theorem: a circulation around a closed curve as a double integral of ∂Q/∂x − ∂P/∂y, and areas from line integrals. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x y\right) dx + \left(2 x\right) dy \), where \( \displaystyle C \) is the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (1, 0) \), \( \displaystyle (0, 1) \), oriented counterclockwise.
Use Green's theorem to find the area enclosed by the ellipse \( \displaystyle \frac{x^2}{16} + \frac{y^2}{16} = 1 \).
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x y\right) dx + \left(x^{3}\right) dy \), where \( \displaystyle C \) is the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (2, 0) \), \( \displaystyle (0, 2) \), oriented counterclockwise.
Use Green's theorem to find the area enclosed by the ellipse \( \displaystyle \frac{x^2}{16} + \frac{y^2}{4} = 1 \).
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x^{2} y\right) dx + \left(2 x\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), oriented counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(- y^{3}\right) dx + \left(2 x\right) dy \), where \( \displaystyle C \) is the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (2, 0) \), \( \displaystyle (0, 2) \), oriented counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x^{2} y\right) dx + \left(x^{3}\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), oriented counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x y\right) dx + \left(x^{2}\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), oriented counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x y\right) dx + \left(3 x\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), oriented counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(y^{2}\right) dx + \left(2 x\right) dy \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 4 \), oriented counterclockwise.