Calculus 12 the definite integral
Specific Curriculum Outcomes
D1 Apply and understand how Riemann’s sum can be used to determine the area under a polynomial curve
D2 Demonstrate an understanding of the meaning of area under the curve
D3 Express the area under the curve as a definite integral
D4 Compute the area under a curve using a numerical integration procedure
C7 Solve initial value problems of the form dy/dx = f(x), y0 = f(x0)
B18 Apply rules for definite integrals
B19 Apply the Fundamental Theorem of Calculus
C8 Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus
C9 Construct antiderivatives using the Fundamental Theorem of Calculus
C10 Find antiderivatives of polynomials, ekx, and selected trigonometric functions of kx
B20 Compute indefinite and definite integrals by the method of substitution
B21 Apply integration by parts to evaluate indefinite and definite integrals (optional)
B22 Solve problems in which a rate is integrated to find the net change over time
D5 Apply integration to calculate areas of regions in a plane
D1 Apply and understand how Riemann’s sum can be used to determine the area under a polynomial curve
D2 Demonstrate an understanding of the meaning of area under the curve
D3 Express the area under the curve as a definite integral
D4 Compute the area under a curve using a numerical integration procedure
C7 Solve initial value problems of the form dy/dx = f(x), y0 = f(x0)
B18 Apply rules for definite integrals
B19 Apply the Fundamental Theorem of Calculus
C8 Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus
C9 Construct antiderivatives using the Fundamental Theorem of Calculus
C10 Find antiderivatives of polynomials, ekx, and selected trigonometric functions of kx
B20 Compute indefinite and definite integrals by the method of substitution
B21 Apply integration by parts to evaluate indefinite and definite integrals (optional)
B22 Solve problems in which a rate is integrated to find the net change over time
D5 Apply integration to calculate areas of regions in a plane
activities

 Math Market Integration Review  Student buy questions from the market and sell solutions back for a profit. A fun activity to review of a variety of techniques of integration including substitution and integration by parts.
 Six Pictures, Four Concepts from Jonathan Claydon  Jonathan tackles the abstract idea of integration and computing areas geometrically using six carefully chosen pictures of graphs.
 Rowing Machine Areas  Ask students to record data from a rowing machine (this could also be done with other exercise equipment such as a treadmill or stationary bicycle). Record the speed every 10 seconds for 2 minutes. Then graph speedtime and calculate the distance using Riemann sums. Students can check to see if they calculated the same answer as the digital readout gave for distance.

 Estimating Areas from Heather Kohn  Heather asks "Which shape has the bigger area? If you were a calculus student, which tool would you choose to explore this?" She used this activity to talk about "What is Calculus?" and ideas of accumulation and integration. Would quarters or dimes be a better/more accurate tool? What about really small circles like beads? As the item gets smaller (towards infinity), you can get a more accurate area. This idea was based on Which One Has a Bigger Area? from Mark Chubb.

