Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only)

The endpoints determine the average radius of the curves

The endpoints affect the sign of the area between the curves

The endpoints define the limits of integration for each curve

The area between the curves is only valid within the endpoints of integration

Setting up the double integral with polar coordinates for the region

Calculating the area bounded by one curve then the other without regard for the intersection points

Determining the limits of integration by finding the angles where the curves intersect

Evaluating the definite integral of the magnitude of force along the path

Determine points of intersection between $r = f(\theta)$ and $r = g(\theta)$.

Calculate the definite integral of $f(\theta) - g(\theta)$ from $\theta=0$ to $\theta=2\pi$.

Find the derivatives of both functions to identify maximum radii.

Integrate both functions separately over their domains.

Because integrals require absolute values for convergence.

To ensure all radii are non-negative values when squaring them.

Because all angles must be converted into radians before integration.

To account for multiplication or division inside an integral.

Multiply $r=e^\theta$ area by two minus $\ln(\theta)$ area considering symmetry without intersection analysis.

Use definite integral from $\pi/4$ to $e$ approximating underestimated area since functions vary less near ends of range.

Approximate integrally using numerical methods such as Simpson's rule due to no exact antiderivative solution available.

Set up an integral $\int_{1}^{e} |e^\theta - \ln(\theta)| d\theta$ assuming polar areas can be directly subtracted like Cartesian ones.

$\int_{0}^{\frac{\pi}{2}} (3\sin(\theta) - 2) d\theta$

$\int_{0}^{\frac{\pi}{2}} \frac{1}{2}(3\sin(\theta))^2 d\theta - \int_{0}^{\frac{\pi}{2}} \frac{1}{2}(2)^2 d\theta$

$\int_{0}^{3} (3\sin(\theta) - 2) r dr$

$\int_{3\sin(0)}^{3\sin(\frac{\pi}{2})} (r - 2) dr$

Only at $\frac{\pi}{4}$ for any integer value $k$.

Wherever $\frac{\theta}{k}$ equals a multiple of $\frac{\pi}{4}$ for any nonzero integer $k$.

At multiples of $\frac{\pi}{8} + k\pi$, where $k$ is an integer.

At multiples of $\frac{\pi}{4} + k\pi$, where $k$ is an integer.

Calculate separate integrals over [$\alpha$, $\beta$] for each curve and subtract smaller areas from larger ones.

Integrate both functions together from angle $\alpha$ to angle $\beta$ and sum up their areas.

Find average radius value between angles alpha ($\alpha$) and beta ($\beta$), then multiply it by $(\beta - \alpha)/2$.

Multiply their maximum radii by half and then integrate from angle $\alpha$ to angle $\beta$.

$r^2 \sin(\theta)$

$\frac{1}{2} r^2 \theta$

$\pi r \theta$

$2 \pi r \theta$

It ensures conditional convergence for all values of $p$ less than one but greater than zero.

It confirms convergence only for $p > 1$.

The test is inconclusive for p-series.

It confirms divergence for all values of $p$ greater than zero.