Area Under a Curve: Quick Revision

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

Strips and the integral

• Strip sum: with $n$ strips of width $\Delta x = \dfrac{b - a}{n}$, the area is estimated by $\sum f(x_i)\,\Delta x$.
• The limit: as $n \to \infty$ the sum tends to $\displaystyle\int_a^b f(x)\,dx$ when $f \ge 0$.
• Shortcut: $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$, where $F' = f$. The constant $C$ cancels.

Which strips

• Under a curve, between $x = a$ and $x = b$: $A = \displaystyle\int_a^b y\,dx$. The vertical lines are the ordinates in Indian books.
• Beside the $y$-axis, between $y = c$ and $y = d$: $A = \displaystyle\int_c^d x\,dy$.
• Circle: $x^2 + y^2 = a^2$ has area $\pi a^2$.
• Ellipse: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ has area $\pi ab$.

Curves that cross the axis

• Signed area: the integral counts a part below the axis as negative. AP and IB call it the net area.
• Total area: $\displaystyle\int_a^b \lvert f(x) \rvert\,dx$. Split at every zero of $f$ inside $(a, b)$, and add the absolute values.
• No limits given: the limits are the points where the curve meets the axis.

A worked example

Find the signed area and the total area of $y = x^2 - 4x$ from $x = 0$ to $x = 6$.

1. The zeros. $x(x - 4) = 0$ at $x = 0$ and $x = 4$. The zero $x = 4$ is inside the interval.
2. The antiderivative. $F(x) = \dfrac{x^3}{3} - 2x^2$. Then $F(0) = 0$, $F(4) = -\dfrac{32}{3}$ and $F(6) = 0$.
3. The pieces. From $0$ to $4$ the value is $-\dfrac{32}{3}$, below the axis. From $4$ to $6$ it is $\dfrac{32}{3}$, above.
4. The answers. The signed area is $0$. The total area is $\dfrac{32}{3} + \dfrac{32}{3} = \dfrac{64}{3}$.

Practice with new numbers

A program to try

The program adds thin strips for the curve of the worked example. It adds the heights with their signs, and it adds their absolute values.

Mistakes to avoid

• Integrating across a zero. The parts cancel. Split the interval at every zero.
• Mixing the strips. Horizontal strips need $dy$ and limits in $y$.
• Forgetting $\pi$ for a circle or an ellipse.
• Adding $+ C$ in a definite integral. It cancels.

Check yourself

1. Find the area under $y = 3x$ from $x = 0$ to $x = 4$.

2. Find the area of the ellipse with semi-axes $6$ and $2$.

3. Find the area between the curve $x = y^2$, the $y$-axis and the lines $y = 0$ and $y = 3$.

4. Find the signed area and the total area of $y = x^3$ from $x = -1$ to $x = 1$.

5. Is the integral of a function always the area between its graph and the $x$-axis?

Sources

• OpenStax, Calculus Volume 1, Chapter 5, sections on approximating areas and the definite integral, and Chapter 6, section on areas between curves.
• NCERT, Mathematics Part II for Class XII, Chapter 8, section on the area under simple curves.