Common Core · AP Calculus AB
FUN-6.D
AP Calculus AB, Unit 6 — For integrands requiring substitution or rearrangement into equivalent forms, determine indefinite integrals and evaluate definite integrals.
A note on this mapping: The Common Core has no calculus standards; this is the AP Calculus AB learning objective.
The 2 skills that cover it
- Applications of IntegralsPut the definite integral to work: the area between two curves, the average value of a function, the derivative of an accumulation function, displacement against distance, substitution for a composite integrand, and total change from a rate in context.Read the lesson →
- Integration by SubstitutionUndo the chain rule: choose u, adjust the constant, integrate powers of an inside function, trigonometric, exponential and logarithm composites and radicals, change the limits of a definite integral, and put x back in terms of u when one is left over.Read the lesson →
What a student should be able to do
- Find the area between two curves, and the area against the axis where a curve dips below.
- Find the average value of a function on an interval.
- Differentiate an accumulation function with the first part of the fundamental theorem.
- Find displacement, distance and net change with an integral, using substitution where the integrand asks for it.
- Choose the substitution u, find du, and work out the constant factor it leaves outside.
- Integrate (ax + b)n, x(x2 + c)n, cos(kx), x ex^2, sqrt(ax + b) and similar composites.
- Recognise the forms that integrate to a logarithm.
- Evaluate a definite integral by changing the limits along with the variable, and rearrange u = x + c into x = u - c when a lone x survives.