Area Between a Curve & a Line

Area Between Boundaries: The area between a curve $y = f(x)$ and a line $y = g(x)$ refers to the finite region completely enclosed by their intersection points. Unlike the area under a single curve, which is bounded by the x-axis, this region is bounded on all sides by the two functions themselves.

Intersection Points as Limits: The x-coordinates where the curve and line meet serve as the lower limit ($a$) and upper limit ($b$) for the definite integral. These points define the horizontal extent of the region being measured.

Vertical Distance: Conceptually, the area is found by integrating the vertical distance between the two functions, calculated as $y_{top} - y_{bottom}$, across the interval $[a, b]$.

Individual Areas: This method involves calculating the area under the curve ($A_C$) and the area under the line ($A_L$) separately using the same limits of integration. Each area is found by integrating the respective function from $a$ to $b$.

Subtraction of Totals: Once both areas are found, the enclosed area is determined by subtracting the smaller area from the larger area. For example, if the curve is above the line, the formula is $Area = A_C - A_L$.

Geometry Shortcuts: If the boundary is a straight line, the area under it often forms a simple geometric shape like a rectangle, triangle, or trapezium. In these cases, using standard geometric formulas (e.g., $\frac{1}{2}(a+b)h$) is often faster and less prone to error than integration.

Combined Integrand: A more efficient approach is to subtract the functions algebraically before integrating. By creating a new function $h(x) = f(x) - g(x)$, you only need to perform a single integration process.

The General Formula: The area is defined by the integral $Area = \int_{a}^{b} [f(x) - g(x)] dx$, where $f(x)$ is the upper function and $g(x)$ is the lower function. This method simplifies the arithmetic by combining like terms before applying the power rule of integration.

Handling Negative Results: If the result of the combined integral is negative, it indicates that the functions were subtracted in the wrong order ($bottom - top$). Since area must be positive, the absolute value (modulus) of the result should be taken.

The 'Top Minus Bottom' Rule: Always identify which function is higher on the y-axis within the limits. Integrating $(lower - upper)$ will result in a negative value, which is a common reason for losing marks if not corrected.

Exact Values vs. Decimals: Exam questions often ask for 'exact area.' This means you should leave your answer as a fraction or in terms of $\pi$ or $\sqrt{n}$ rather than providing a rounded decimal from a calculator.

Check the Limits: A common mistake is using the y-intercepts or x-intercepts of the individual functions as limits. Always use the $x$-coordinates of the intersections between the two functions.

Calculator Verification: If allowed, use the definite integral function on your calculator to verify your manual integration. This provides a 'sanity check' for your final numerical answer.