The Area-Under-a-Curve Idea

Until now, the definite integral $\int_a^b f(x)\,dx$ has mostly been a number you compute by finding an antiderivative and applying the Fundamental Theorem of Calculus. This chapter gives that number a geometric meaning: when $f(x) \geq 0$ throughout $[a,b]$, the definite integral $\int_a^b f(x)\,dx$ equals the exact area of the region bounded above by the curve $y=f(x)$, below by the $x$-axis, and on the sides by the vertical lines (ordinates) $x=a$ and $x=b$.

Why the definite integral is an area -- the Riemann sum picture. Split the interval $[a,b]$ into $n$ equal strips, each of width $\Delta x = \dfrac{b-a}{n}$. On the $i$-th strip, erect a thin rectangle of height $f(x_i)$, where $x_i$ is a sample point in that strip. The area of this one rectangle is $f(x_i)\,\Delta x$, and adding up all $n$ of them gives an approximation to the area under the curve:

This is only an approximation because the tops of the rectangles are flat while the curve is (generally) not -- some rectangles overshoot the curve slightly, others undershoot it. As $n \to \infty$, each strip becomes infinitesimally thin ($\Delta x \to 0$), the flat-topped rectangles hug the curve more and more closely, and the error in the approximation shrinks to zero. This limit is defined to be the definite integral:

Since this limit is precisely the limiting value of a sum of rectangular-strip areas that increasingly fills up the region under the curve, the definite integral is the area of that region -- not merely a number that happens to equal it.

The role of the Fundamental Theorem of Calculus. Evaluating the Riemann-sum limit directly is usually impractical, which is exactly why the Fundamental Theorem of Calculus is so useful here: if $F(x)$ is any antiderivative of $f(x)$ (that is, $F'(x) = f(x)$), then $\int_a^b f(x)\,dx = F(b) - F(a)$. Every area calculated in this chapter uses this shortcut -- find an antiderivative, then subtract its value at the lower limit from its value at the upper limit -- while relying on the Riemann-sum argument above to justify why the resulting number is a genuine area.

Sign convention. The formula $\int_a^b f(x)\,dx$ gives a positive area only when $f(x) \geq 0$ on $[a,b]$, matching every curve considered in this chapter (a line segment kept above the $x$-axis, the upper half of a circle or ellipse, or a parabola's arm). If a curve dips below the $x$-axis, the integral there comes out negative (since each strip's "height" $f(x_i)$ is negative), and the geometric area is recovered by taking the absolute value of that piece before adding it to the rest -- area can never be negative, even though the raw integral can be.

Extending to two curves. The very same idea -- summing thin strips and taking the limit -- extends directly to the area between two curves $y = f(x)$ (the upper one) and $y = g(x)$ (the lower one) over an interval where $f(x) \geq g(x)$: each representative strip now has height $f(x_i) - g(x_i)$ instead of just $f(x_i)$, giving

This single working principle -- identify the region, draw a representative thin strip, write its area as (height)$\times dx$ (or, when it is more convenient, (width)$\times dy$), and integrate over the range the strip sweeps through -- is what every remaining section of this chapter applies to a straight line, a circle, a parabola, an ellipse, and finally to the region enclosed between two of these curves at once.