Applications of Integrals

1. Area under a curve

For curve y = f(x), bounded by x-axis and ordinates x = a and x = b:

A = ∫_a^b f(x) dx

If the area lies below the x-axis, take the magnitude of the integral.

2. Area under curve x = f(y)

Between the y-axis and the lines y = c and y = d:

A = ∫_c^d x dy

3. Area between two curves

For curves y = f(x) and y = g(x) between x = a and x = b:

A = ∫_a^b [f(x) – g(x)] dx

4. Average value of a function

For y = f(x) over the interval [a, b]:

y_avg = (1 / (b – a)) ∫_a^b f(x) dx

5. Curve Tracing (Steps)

• Check symmetry (x-axis, y-axis, line y = x, opposite quadrants).
• Find points where dy/dx = 0 (horizontal tangents).
• Find the intercepts on the x-axis and y-axis.
• Check intervals where f(x) increases/decreases.
• Examine behaviour as x → ∞ or –∞.

6. Useful Results

• Area of ellipse (x²/a² + y²/b² = 1): A = πab
• Area between parabolas y² = 4ax and x² = 4by: A = 16ab/3
• Area between parabola y² = 4ax and line y = mx: A = 8a² / (3m³)