Application of Integrals — chapter notes

If $f$ is continuous and $f(x) \ge 0$ on $[a, b]$, the area of the region bounded by $y = f(x)$, the $x$-axis and the ordinates $x = a$ and $x = b$ is $\displaystyle\int_a^b f(x)\,dx$.

Think of thin vertical strips of width $dx$ and height $f(x)$; the integral adds them up.

Always sketch the region first and mark the limits.

If $f(x) \le 0$ on $[a, b]$, the area is $-\displaystyle\int_a^b f(x)\,dx$.

If the curve crosses the axis inside $[a, b]$, find its zeros, split the interval there, and add the absolute values of the pieces. Integrating straight across a zero lets the pieces cancel and gives a wrong (too small, or even zero) answer: for example $\displaystyle\int_0^{2\pi} \sin x\,dx = 0$, but the area is $4$.

• Horizontal strips: for $x = g(y) \ge 0$ between $y = c$ and $y = d$, area $= \displaystyle\int_c^d g(y)\,dy$. In general, length of strip $=$ right $x$ $-$ left $x$.
• Modulus: split at the corner where the inside is zero; each piece is a line, so the region is made of triangles or trapeziums.
• Exponential: $e^x > 0$ and $a^x > 0$, so the curve is always above the axis; $\displaystyle\int a^x\,dx = \dfrac{a^x}{\log a}$.
• Logarithmic: $\log x < 0$ on $(0, 1)$; $\displaystyle\int \log x\,dx = x\log x - x$.

For $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, the upper half is $y = \dfrac{b}{a}\sqrt{a^2 - x^2}$. By symmetry about both axes,

Area $= 4 \cdot \dfrac{b}{a}\displaystyle\int_0^a \sqrt{a^2 - x^2}\,dx = 4 \cdot \dfrac{b}{a} \cdot \dfrac{\pi a^2}{4} = \pi ab$.

A circle of radius $r$ is the case $a = b = r$, giving $\pi r^2$. Parts of a circle (between two ordinates) use the standard integral directly.

If $f(x) \ge g(x)$ on $[a, b]$, the area between the curves is $\displaystyle\int_a^b \big(f(x) - g(x)\big)\,dx$; this holds even if part of the region is below the $x$-axis.

Steps: solve $f(x) = g(x)$ for the meeting points; test one point to decide which curve is upper; integrate upper minus lower. If the curves cross inside the interval, split there and take upper minus lower on each part.

• A segment of a circle or ellipse cut off by a chord is the area under the arc minus the area under the chord.
• When the boundary of the region changes from one curve to another (for example the region inside both a circle and a parabola), split at the meeting point and integrate each part.
• Use symmetry: if both curves are even, find the area for $x \ge 0$ and double it; for a region symmetric about the $x$-axis, double the upper half.
• Regions described by inequalities, such as $\{(x, y) : x^2 \le y \le |x|\}$, are read as "lower curve $\le y \le$ upper curve".