Compute, in terms of $A, B, h$, and $k$, the area enclosed by the curve defined by the parametric equations:
$x(θ)=Acosθ+h$
$y(θ)=Bsinθ+k \quad \quad \quad \quad \quad \quad $ for $0 ≤ \theta ≤ 2π$.
Since those equations obviously form an ellipse, I first tried using the formula $Area_{ellipse} = \pi r_1r_2$ and ended up with $AB \pi$, and that was apparently wrong.
Next, I tried to use the formula $\int_a^bg(t)f^\prime(t)dt$, ending up with the integral $\int_0^{2\pi}-AB*sin^2\theta-Aksin\theta \quad d\theta$ and the answer $-AB \pi$, which is also wrong.
Is there are formular or something that I'm missing?
Thanks for your help!
Best Answer
The provided equations are
$$x\left(\theta\right) = A \cos\theta + h \tag{1}\label{eq1}$$ $$y\left(\theta\right) = B \sin\theta + k \tag{2}\label{eq2}$$
where $0 ≤ \theta ≤ 2\pi$. As you've already correctly surmised, this is an equation of an ellipse, determined by, in \eqref{eq1}, subtracting $h$ & dividing by $A$ and, in \eqref{eq2}, subtracting $k$ & dividing by $B$, to get
$$\cos\theta = \frac{x\left(\theta\right) - h}{A} \tag{3}\label{eq3}$$ $$\sin\theta = \frac{y\left(\theta\right) - k}{B} \tag{4}\label{eq4}$$
Squaring both sides of each & adding, using that $\sin^2\theta + \cos^2\theta = 1$, gives
$$ \left(\frac{x\left(\theta\right) - h}{A}\right)^2 + \left(\frac{y\left(\theta\right) - k}{B}\right)^2 = 1 \tag{5}\label{eq5}$$
As can be seen, such as indicated in Wikipedia's Ellipse, this is a general equation of an ellipse, with a center at $\left(h,k\right)$ and semi-major & semi-minor axes of $\left|A\right|$ and $\left|B\right|$. Note that since the axes are always non-negative values, and there was no stipulation that $A$ and $B$ are both positive, it's important to account for this issue by taking absolute values.
As for the area, it will thus be $\left|AB\right|\pi$, using absolute value signs here again since the area is always non-negative. One final thing to note is that if $A$ or $B$ are $0$, then \eqref{eq3} to \eqref{eq5} are invalid. Nonetheless, the answer for the area would still be valid. This is because $x\left(\theta\right)$ and/or $y\left(\theta\right)$ would just be a constant value, thus giving an "ellipse" with no width and/or height, so the resulting area would be $0$.