Preview Activity 12.8.1.
(a)
Consider the double integral
\begin{equation}
I = \iint_D x^2+y^2 \, dA \tag{12.8.1}
\end{equation}
where \(D\) is the upper half of the unit disk (\(y \geq 0\)). Remember the polar conversion equations:
\begin{equation}
x = r \cos(\theta) \quad \text{ and } \quad y = r \sin(\theta)\tag{12.8.2}
\end{equation}
(a)
Write the double integral \(I\) given in Equation (12.8.1) as an iterated integral in rectangular coordinates.
(b)
Write the double integral \(I\) given in Equation (12.8.1) as an iterated integral in polar coordinates.
(c)
Draw a plot of the region of integration \(D\) in the \(xy\)-plane and label the four “corners” of the polar coordinates (if \(D\) is described by \(a \leq r \leq b , c \leq \theta \leq d\text{,}\) then the corners will be \((r,\theta)=(a,c),(a,d),(b,c),(b,d)\)).
(b)
Suppose you are calibrating a new radar system for your self driving car. This radar system is mounted at the center of your front bumper and can detect objects up to 45 degrees to either side of the forward direction at distances from 1 meter to 3 meters.
(a)
On Figure 12.8.1, draw a plot of \(S\text{,}\) the region that your radar system can detect objects. Assume your car’s front bumper is centered at the origin of your coordinate system and the car is facing the positive \(y\)-direction. You should label the four corners of \(S\) in terms of both rectangular and polar coordinates.
(b)
On Figure 12.8.2, plot the rectangle on the \(r\theta\)-plane that corresponds to the bounds \(a \leq r \leq b \quad c \leq \theta \leq d\) for your radar system. Label the four corners of your figure with their \(r\) and \(\theta\) coordinates.
