Find the area bounded by the arc of the curve (cardioid)

Find the area bounded by the arc of the curve (cardioid): $\rho = a(1 + \cos\varphi)$.

We will apply the polar-coordinates area formula over the full curve.

Step 1: Understand the cardioid

At $\varphi=0$: $\rho=2a$ (furthest point). At $\varphi=\pi$: $\rho=0$ (cusp at the origin). The curve is symmetric about the polar axis.

Step 2: Set up the polar area formula

\[A = \frac{1}{2}\int_0^{2\pi} [a(1+\cos\varphi)]^2\,d\varphi\]

Step 3: Expand the integrand

\[[a(1+\cos\varphi)]^2 = a^2\left(1+2\cos\varphi+\cos^2\varphi\right) = a^2\left(\frac32+2\cos\varphi+\frac{\cos2\varphi}{2}\right)\]

Step 4: Integrate

\[A = \frac{a^2}{2}\left[\frac{3\varphi}{2}+2\sin\varphi+\frac{\sin2\varphi}{4}\right]_0^{2\pi} = \frac{a^2}{2}(3\pi) = \frac{3\pi a^2}{2}\]

\[\boxed{A = \frac{3\pi a^2}{2} \text{ square units}}\]

Step 5: Comparison with a circle

A circle of radius $a$ has area $\pi a^2$; the cardioid encloses $1.5$ times that area.

Step 6: Numerical check

For $a=1$: $A=\frac{3\pi}{2}\approx4.712$, versus a unit circle's $\pi\approx3.142$; indeed $4.712\approx1.5\times3.142$. ✓

Summary

\[\boxed{A = \frac{3\pi a^2}{2} \text{ square units}}\]

Key insight: The heart-shaped cardioid encloses exactly $\frac32$ times the area of a circle with the same parameter $a$.