Integral Calculus and Its Applications
Integral Calculus and Its Applications tests definite and indefinite integration, standard techniques, and the use of integrals for area and volume. The exam often combines a substitution or integration by parts with a definite limit.
Techniques and applications
Use substitution when an integrand contains a function and its derivative; change the limits at the same time for a definite integral. Integration by parts: ∫u dv=uv−∫v du; choose u by the usual priority (inverse trig, log, algebraic, trig, exponential). Partial fractions handle rational functions. For area, integrate |y| between the appropriate x-limits; for volume about the x-axis, use V=π∫y² dx. Properties include ∫a^b f=−∫b^a f and additivity over adjacent intervals.
Worked example
Evaluate ∫0^1 x e^x dx. Let u=x, dv=e^x dx; then du=dx, v=e^x. So the integral is [x e^x]0^1−∫0^1 e^x dx = (1·e−0)−[e^x]0^1 = e−(e−1)=1.
Traps
When substituting in a definite integral, either change the limits or convert back to x before evaluating; mixing both is a common error. Area below the x-axis needs an absolute value or a sign split. The volume formula uses y², not y. In integration by parts, keep the sign of the subtracted integral. Do not omit +C for indefinite integrals.