Chapter 9: Integration

Indefinite integration recovers an original function from its derivative, with the power rule serving as a primary computational method for polynomial expressions. The rule applies to rational exponents and extends to composite linear functions, while the constant of integration must accompany all indefinite integrals since the derivative of any constant equals zero. Integration obeys linearity properties, allowing constant factors to be extracted and sums of functions to be integrated term by term. Definite integration evaluates an antiderivative between specified upper and lower bounds, producing a unique numerical result independent of the constant of integration. The fundamental theorem of calculus connects this computational process directly to the geometric concept of area under a curve. Several key properties govern definite integrals, including the reversal of limits that negates the result and the ability to partition integrals across adjacent intervals. Computing areas bounded by curves and coordinate axes relies on setting up appropriate definite integrals, with special attention required when regions extend below the x-axis since the integral yields negative values in such cases. Areas between multiple curves are found by integrating the absolute difference of the bounding functions. Improper integrals extend the theory to scenarios with infinite limits or discontinuities by replacing unbounded values with limits of finite expressions. The chapter concludes with volumes of revolution, where rotating a planar region about an axis creates a three-dimensional solid whose volume is computed using the disk method with circular cross-sections perpendicular to the axis of rotation.