Area Under a Curve

The central idea

For a continuous function $f\ge0$ on $[a,b]$, the geometric area between $y=f(x)$ and the $x$-axis is $A=\int_a^b f(x)\,dx$. If $f$ changes sign, the definite integral is signed net area; total geometric area is $\int_a^b|f(x)|\,dx$, usually evaluated by splitting at the zeros of $f$.

Definitions, hypotheses, and notation

The phrase 'under the curve' should be parsed carefully. If the graph lies below the axis, the region is geometrically between the curve and the axis, but the integral contributes a negative signed amount. If the curve crosses repeatedly, the zeros divide the interval into pieces on which $|f|$ is either $f$ or $-f$. Computing those pieces is equivalent to integrating $|f|$, and it avoids the false shortcut $|\int f|$.

Units provide a strong diagnostic. If $x$ is measured in seconds and $f(x)$ in metres per second, the integral has metres, so it represents displacement rather than planar square metres. Only when both axes represent compatible lengths is the numerical integral literally a geometric area in square units. The same signed-accumulation mathematics supports both interpretations; the context decides the name and units.

Conceptual meaning

A definite integral is the limit of thin rectangle areas. Positive heights add area above the axis and negative heights subtract area below it. This is why displacement can be negative while total distance cannot.

A dependable method and decision rule

1. Sketch or analyze the graph and identify the interval.
2. Solve $f(x)=0$ inside the interval to locate sign changes.
3. Choose net area or geometric area according to the wording.
4. Integrate on each sign-consistent subinterval using the Fundamental Theorem.
5. For geometric area, make each contribution nonnegative before adding.

Fully worked example

Graphical or geometric meaning

The two pieces are triangles: one has base and height $1$, area $1/2$; the other has base and height $2$, area $2$. This geometric check exposes the difference between total area $5/2$ and signed area $3/2$.

Common mistakes and why they fail

Verification and reasonableness checks

• Estimate area from a sketch before integrating.
• Verify that every geometric-area contribution is nonnegative.
• Confirm that the answer has squared units when the axes have length units.

Signed integral versus geometric area

A definite integral records signed accumulation, so a graph below the axis contributes negatively. Geometric area instead requires splitting at every intercept and integrating $|f|$, or reversing the sign on negative subintervals. A quick sketch should precede the calculation: it reveals sign changes, gives a plausible scale, and prevents cancellation from being mistaken for a small region. After evaluating, compare with a bounding rectangle. If a nonnegative graph has height at most $M$ over an interval of length $L$, its area must lie between $0$ and $ML$. This bound and the sign pattern offer independent checks on an antiderivative computation.

Practice

1. Find the area under $y=2x$ on $[0,2]$.
2. Find the net area of $y=x-2$ on $[0,4]$.
3. Find its total geometric area on $[0,4]$.

Connections and next steps

• Riemann Sums
• Definite Integral
• Area Between Curves