The fundamental theorem of calculus, accumulation functions and properties of definite integrals

Learning objectives

Define an accumulation function g(x) = ∫ₐˣ f(t) dt and apply the first part of the fundamental theorem of calculus to find its derivative, including when the upper limit is a function of x, using the chain rule

Analyze an accumulation function from the graph of its integrand: evaluate it using the areas of geometric shapes, find where it is increasing or decreasing, locate its extrema and inflection points, and sketch its graph

Apply the properties of definite integrals: reversing the limits, an integral over a single point, splitting an interval, constant multiples, sums and differences, and the effect of symmetry

Evaluate a definite integral using the second part of the fundamental theorem of calculus, F(b) − F(a), and use it to express a function value as an initial value plus the integral of its derivative

Evaluate definite integrals of functions given by graphs made of line segments and circular arcs, and of functions given by tables, combining geometry with the properties of integrals

Use a graphing calculator to evaluate definite integrals in calculator-active questions and report answers to the required accuracy