Key Points

The fundamental concept is that the area of a region bounded by curves can be precisely calculated by evaluating a definite integral. This method effectively sums the areas of an infinite number of infinitesimally thin rectangular strips.

The area bounded by a curve $y=f(x)$, the x-axis, and the vertical lines (ordinates) $x=a$ and $x=b$ is given by the formula $A = \int_{a}^{b} y \, dx = \int_{a}^{b} f(x) \, dx$. This method uses vertical strips of width $dx$.

The area bounded by a curve $x=g(y)$, the y-axis, and the horizontal lines $y=c$ and $y=d$ is given by the formula $A = \int_{c}^{d} x \, dy = \int_{c}^{d} g(y) \, dy$. This method uses horizontal strips of width $dy$.

Area is always a non-negative quantity. If the curve $y=f(x)$ lies below the x-axis over the interval $[a, b]$, the definite integral will be negative. The required area is the absolute value: $A = \left| \int_{a}^{b} f(x) \, dx \right|$.

If a curve crosses the x-axis at $x=c$ within the interval $[a, b]$, you must split the integral. The total area is the sum of the absolute values of the separate integrals: $A = \left| \int_{a}^{c} f(x) \, dx \right| + \left| \int_{c}^{b} f(x) \, dx \right|$.

The area between two curves $y=f(x)$ (upper curve) and $y=g(x)$ (lower curve) from $x=a$ to $x=b$ is given by $A = \int_{a}^{b} [f(x) - g(x)] \, dx$. Ensure $f(x) \geq g(x)$ for all $x$ in $[a, b]$.

The area between two curves $x=f(y)$ (right curve) and $x=g(y)$ (left curve) from $y=c$ to $y=d$ is given by $A = \int_{c}^{d} [f(y) - g(y)] \, dy$. Ensure $f(y) \geq g(y)$ for all $y$ in $[c, d]$.

If the area is symmetric about an axis or the origin, calculate the area of one symmetric portion and multiply by the number of such portions. For example, the area of a circle is 4 times the area in the first quadrant.

The area enclosed by the circle $x^2 + y^2 = a^2$ is $\pi a^2$. This is derived by calculating $A = 4 \int_{0}^{a} \sqrt{a^2 - x^2} \, dx$.

The area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $\pi ab$. This is derived by calculating $A = 4 \int_{0}^{a} \frac{b}{a} \sqrt{a^2 - x^2} \, dx$.

The integral $\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C$ is frequently used for calculating the areas of circles and ellipses.

To find the area of a region: 1. Sketch the curves and identify the bounded region. 2. Find the points of intersection to determine the limits of integration. 3. Set up the appropriate definite integral (upper curve - lower curve) and evaluate it.