Application of Integrals: Tricky Questions Solved (1)

Question

Find the area enclosed by the curve $y = x^2$, the $x$-axis, and the lines $x = 1$ and $x = 3$.

Solution — Step by Step

Why This Works

A definite integral computes the signed area between a curve and the $x$-axis. When the curve is above the axis (positive $y$), the integral gives a positive area directly. When the curve dips below, we get a negative contribution. For unsigned area, we integrate $|f(x)|$ or split the interval at zeros.

For $y = x^2$ on $[1, 3]$, the curve is above the axis throughout, so the integral gives the geometric area straight up.

Alternative Method

Use the average value approach. The average of $x^2$ on $[1, 3]$ is approximately $(1 + 9)/2 = 5$ (rough Simpson estimate). Multiplying by interval length $2$ gives $\approx 10$, close to the exact $26/3 \approx 8.67$. Useful as a sanity check.

For a cleaner numerical method, the function is monotonic, so the area is between rectangles of heights $1$ and $9$ over width $2$: between $2$ and $18$. The exact value $8.67$ fits.

Common Mistake

Forgetting to take absolute value when the curve dips below the axis. The phrase “area enclosed” usually means geometric (positive) area. If the integral gives a negative number, take its magnitude.