Area Between Curves: Quick Revision

A brief note on Area Between Curves. Read it before a test. Try each question at the end before you open its answer.

The formula

• Area between two curves: $\displaystyle\int_a^b \big(f(x) - g(x)\big)\,dx$, where $f$ is above $g$ on $[a, b]$.
• Limits: the solutions of $f(x) = g(x)$. Sketch first.
• Which is above: put one $x$ between the limits into both formulas.
• Position does not matter: the formula holds above, below or across the $x$-axis.

Shortcuts

• Parabola cut by a line: area $= \dfrac{|a|\,d^3}{6}$, where $a$ is the coefficient of $x^2$ and $d$ is the distance between the meeting points.
• Two parabolas: use the difference of the two $x^2$ coefficients in place of $a$.
• Symmetry: if the region is symmetric about an axis, double one half.

Crossings and strips

• Curves that cross inside the range: split at every crossing and add the areas, or integrate $|f - g|$. One signed integral lets lobes cancel.
• Vertical strips: height is upper minus lower, with $dx$.
• Horizontal strips: width is right minus left, with $dy$ and limits that are $y$-values.
• Choose well: pick the direction in which the top and the bottom of the strip stay the same curve.

A worked example

Find the area between $y = x$ and $y = 2 - x^2$.

1. Meeting points. $x = 2 - x^2$ gives $x^2 + x - 2 = 0$, so $x = -2$ or $x = 1$.
2. Which is above? At $x = 0$ the parabola gives $2$ and the line gives $0$. The parabola is above.
3. The area. $\displaystyle\int_{-2}^{1} (2 - x^2 - x)\,dx = \dfrac{9}{2}$. The shortcut agrees: $\dfrac{1 \times 3^3}{6} = \dfrac{9}{2}$.

Practice with new numbers

A program to try

The program finds the area between $y = x^2$ and $y = x + 6$, and then shows that two lobes cancel in a signed integral.

Mistakes to avoid

• Guessing the limits. Solve $f(x) = g(x)$.
• Subtracting the wrong way. An area is positive. Test one point.
• One integral across a crossing. The lobes cancel. Split the region.
• Wrong direction of strips. If the lower edge of a vertical strip changes, try horizontal strips.

Check yourself

1. Find the area between $y = x^2$ and $y = x + 6$.

2. Find the area enclosed by $y = x^3$ and $y = 2x$.

3. Find the area between $x = y^2$ and $x = 3 - 2y$, using horizontal strips.

4. Why can one signed integral give $0$ for a region that has area?

5. Find the area between $y = \sqrt{x}$ and $y = \dfrac{x}{2}$.

Sources

• OpenStax, Calculus Volume 1, Chapter 6, section 6.1 on areas between curves.
• NCERT, Mathematics Part II for Class XII, Chapter 8, the section on the area between two curves.