Q.The area of the region bounded by the circle $x^2 + y^2 = 1$ is
(A) $2\pi$ sq units
(B) $\pi$ sq units
(C) $3\pi$ sq units
(D) $4\pi$ sq units

The key idea is that the area enclosed by a circle of radius $r$ is $\pi r^2$.

The given equation $x^2 + y^2 = 1$ represents a circle centered at the origin with radius $r = 1$. …

The question asks for the area of the region bounded by the circle $x^2 + y^2 = 1$. This is a standard result from geometry, but let's understand it from the perspective of integration — the "Area Under Curve" approach — so you see why the answer is what it is, not just that it is.

The equation $x^2 + y^2 = 1$ describes a circle centered at the origin with radius $r = 1$. The area of any circle is $\pi r^2$, so here it's $\pi \cdot 1^2 = \pi$. That's the direct answer. But if you were to compute this using calculus, you'd set up an integral for the area between the upper and lower halves of the circle.

1. Solve for $y$ in terms of $x$.

   From $x^2 + y^2 = 1$, we get $y = \pm \sqrt{1 - x^2}$. The upper semicircle is $y = \sqrt{1 - x^2}$, and the lower is $y = -\sqrt{1 - x^2}$. The region is symmetric about the $x$-axis.

2. Set up the area integral.

   The area between two curves $y_{\text{top}}$ and $y_{\text{bottom}}$ from $x = a$ to $x = b$ is $\int_a^b (y_{\text{top}} - y_{\text{bottom}}) \, dx$. Here, $y_{\text{top}} = \sqrt{1 - x^2}$, $y_{\text{bottom}} = -\sqrt{1 - x^2}$, and the circle runs from $x = -1$ to $x = 1$. So:

3. Evaluate the integral. The integral $\int_{-1}^{1} \sqrt{1 - x^2} \, dx$ represents the area of a semicircle of radius 1 (the upper half). That area is $\frac{\pi}{2}$. Multiplying by 2 gives the full circle area: $\text{Area} = 2 \cdot \frac{\pi}{2} = \pi.$ …

Method: Area enclosed by a circle centred at the origin

Use this for the region bounded by a full circle $x^2 + y^2 = r^2$.

Steps

Step 1: Read the radius from the equation.

Writing the circle as $x^2 + y^2 = r^2$, the constant on the right is $r^2$, so $r = \sqrt{\text{that constant}}$.

Step 2: Apply the area formula.

Step 3 (why it works by integration): treat it as area between two semicircles.

Solving for $y$ gives $y = \pm\sqrt{r^2 - x^2}$, so …

Common Mistakes

Mistake 1: Confusing circumference with area.

Why it's wrong: the circumference of $x^2+y^2=1$ is $2\pi r = 2\pi$, which is the trap option (A); the area is a different quantity. Correct approach: use $\text{Area} = \pi r^2 = \pi(1)^2 = \pi$.

Mistake 2: Forgetting to square the radius. …