Integral Calculus and Applications
Core concepts
Integration reverses differentiation. You need indefinite integrals with the constant of integration, definite integrals evaluated via the fundamental theorem, and the area interpretation. Areas between curves are found by integrating the upper minus the lower function over the intersection interval. Volumes of revolution use the disc method, integrating πy² with respect to x.
Substitution reverses the chain rule and is the most common technique; recognising the derivative of the inner function is the key skill. For definite integrals, change the limits when substituting, or convert back before evaluating. Average value of a function and accumulated change problems also appear.
Traps to watch
- Omitting the constant of integration in indefinite integrals.
- Forgetting to change limits when substituting in a definite integral.
- Computing a signed integral when the question asks for total area, so negative regions must be split and their absolute values added.
- Mixing up upper and lower functions when finding the area between curves.
Rule of thumb
Sketch the region first. For area, split at any x-intercepts; for substitution, either change the limits or back-substitute before evaluating.
Worked example
Find the area between y = x² and y = x from x = 0 to x = 1. The line is above the curve on this interval, so integrate (x − x²) from 0 to 1, giving [x²/2 − x³/3] evaluated from 0 to 1, which equals ½ − ⅓ = ⅙.