Area Under Simple Curves

If \(f(x)\ge0\) on \([a,b]\), the area of the region bounded by \(y=f(x)\), the \(x\)-axis and the lines \(x=a,\ x=b\) is

\[A=\int_a^bf(x)\,dx=\int_a^by\,dx.\]

Think of the region as made of thin vertical strips of height \(y\) and width \(dx\); integration adds them up. Similarly, with horizontal strips, the area between \(x=g(y)\), the \(y\)-axis and \(y=c,\ y=d\) is \(\int_c^dx\,dy\).

Regions below the axis

If \(f(x)\le0\), \(\int_a^bf\) is negative; the area is \(\left|\int_a^bf\right|\). If \(f\) changes sign, split the interval at its zeros and add the absolute values.

Standard curves

• Circle \(x^2+y^2=a^2\): \(A=4\int_0^a\sqrt{a^2-x^2}\,dx=\pi a^2\).
• Ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\): \(A=4\cdot\frac ba\int_0^a\sqrt{a^2-x^2}\,dx=\pi ab\).
• Parabola \(y^2=4ax\) and the line \(x=a\) (latus rectum): \(A=2\int_0^a2\sqrt{ax}\,dx=\frac{8a^2}{3}\).

Method

1. Sketch the curve and shade the region.
2. Decide vertical (dx) or horizontal (dy) strips.
3. Write the limits from the sketch; use symmetry.
4. Integrate and state the area in square units.