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Topic 1: Further integration

Fundamental theorem of calculus and definite integrals

3 hours

• Use sums of the form ∑ 𝑓(𝑥𝑖) 𝛿𝑥𝑖𝑖 to estimate the area under the curve 𝑦 = 𝑓(𝑥). 1 interactive
• Recognise the definite integral ∫ 𝑓(𝑥)𝑏 𝑎  𝑑𝑥 as a limit of sums of the form ∑ 𝑓(𝑥𝑖) 𝛿𝑥𝑖𝑖.
• Understand the fundamental theorem of calculus, ∫ 𝑓(𝑥)𝑏 𝑎  𝑑𝑥= 𝐹(𝑏)− 𝐹(𝑎), and use it to calculate definite integrals. 1 interactive
• Use the definite integral ∫ 𝑓(𝑥)𝑏 𝑎  𝑑𝑥 to determine the area under the curve 𝑦 = 𝑓(𝑥) between 𝑥 = 𝑎 and 𝑥 = 𝑏 if 𝑓(𝑥) > 0 over this interval.

Applications of integration

8 hours

• Calculate the area enclosed by a curve and the 𝑥-axis over a given domain, with and without technology.
• Calculate the area between curves, with and without technology.
• Use the trapezoidal rule, ∫ 𝑓(𝑥)𝑏 𝑎  𝑑𝑥 ≈ 𝑤 2 [𝑓(𝑥0)+ 2(𝑓(𝑥1)+ 𝑓(𝑥2)+ 𝑓(𝑥3)+... 𝑓(𝑥𝑛−1))+ 𝑓(𝑥𝑛)], where 𝑤 = 𝑏−𝑎 𝑛, to approximate an area and the value of a definite integral, with and without technology. 1 interactive
• Calculate total change by integrating instantaneous or marginal rates of change, with and without technology.
• Model and solve problems that involve definite integrals, including motion problems, with and without technology.