Integration, area under a curve and the trapezoidal rule

Learning objectives

Understand integration as the reverse of differentiation, and find the antiderivative of a sum of terms ax^n for integer n other than −1, remembering the constant of integration

Determine the constant of integration from a boundary condition, for example finding a cost function from a marginal cost function and a known fixed cost

Evaluate a definite integral with technology, and interpret it as the area between the curve y = f(x) and the x-axis over an interval where f(x) > 0

Interpret a definite integral of a rate of change as the total change accumulated over an interval, and give the units of the result

Approximate the area under a curve using the trapezoidal rule with h = (b − a)/n, adding the two end ordinates and twice each interior ordinate before multiplying by h/2, including the case where the function is known only as a table of measured values such as widths across an irregular region

Decide whether the trapezoidal rule gives an overestimate or an underestimate from the curvature of the graph, and explain how increasing the number of strips improves the approximation