[Steps Shown] Explain why (1/2) ∫_a^b(X(t) Y^prime(t)-Y(t) X^prime(t)) d t measures the area of R provided that (X(t), Y(t)) sweeps out the boundary
Question: Explain why
\[\left(\frac{1}{2}\right) \int_{a}^{b}\left(X(t) Y^{\prime}(t)-Y(t) X^{\prime}(t)\right) d t\]measures the area of \(\mathrm{R}\) provided that \((\mathrm{X}(\mathrm{t}), \mathrm{Y}(\mathrm{t}))\) sweeps out
the boundary of \(\mathrm{R}\) exactly one time in the counterclockwise
fashion as \(\mathrm{t}\) advances from a to \(\mathrm{b}\).
Deliverable: Word Document 
![[Solution Library] Here 's the plot of](/images/solutions/MC-solution-library-37594.jpg "[Solution Library] Here 's the plot of a whimsical region R in the xy-plane.")
[Solution Library] Here 's the plot of a whimsical region R in the xy-plane.
![[Solution Library] Here 's the plot of](/images/solutions/MC-solution-library-37595.jpg "[Solution Library] Here 's the plot of another region R in the x y -plane.")
[Solution Library] Here 's the plot of another region R in the x y -plane.
![[See Solution] Go with f(x)=x e^-x and](/images/solutions/MC-solution-library-37596.jpg "[See Solution] Go with f(x)=x e^-x and explain with words and")
[See Solution] Go with f(x)=x e^-x and explain with words and
![[Solution Library] Go with f(x, y)=2+ sin](/images/solutions/MC-solution-library-37597.jpg "[Solution Library] Go with f(x, y)=2+ sin (y)-x / 2 and explain with")
[Solution Library] Go with f(x, y)=2+ sin (y)-x / 2 and explain with
(Solution Library) Find the average value of cos (x) on
Area and Volume of a Pyramid - MathCracker.com