Domain: Calculate and Apply Integrals Cluster: Apply integrals to solve problems Use a definite integral to find the volume of a formed by rotating around a given axis. Objectives: Students will: Find the volume of a solid of revolution using the disk method. Find the volume of a solid of revolution using the washer method. Find the volume of a with a known cross section. Find the volume of a solid of revolution using the shell method. Compare the uses of the disk method and the shell method. 7.2: Volume: The Disk Method 7.3: Volume: The Shell Method Additional Resource(s) Larson Calculus Videos – Section 7.2 Larson Calculus Videos – Section 7.3 Khan Academy Calculus Videos Section Project (after 7.3) See engageny Lesson for Exit Ticket/Discussion Questions. Domain: Understanding Integrals Cluster: Demonstrate understanding of a Definite Integral Define the definite integral as the of Riemann sums and as the net accumulation of change. Correctly write a Riemann sum that represents the definition of a definite integral. Objectives: Students will: Understand the definition of Riemann sums. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals. 4.3: Riemann Sums and Definite Integrals Additional Resource(s) Larson Calculus Videos – Section 4.3 Khan Academy Calculus Videos Writing in Math/Discussion Give an example of a function that is integrable on the interval [-1, 1], but not continuous on [-1, 1] Domain: Understanding Integrals Cluster: Understand and apply the Fundamental Theorem of Calculus Evaluate definite integrals the Fundamental Theorem of Calculus. Use the Fundamental Theorem of Calculus to represent a particular antiderivative of a function and to Objectives: Students will: Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for integrals. Find the average value of a function over a closed interval. 4.4: The Fundamental Theorem of Calculus Additional Resource(s) Larson Calculus Videos – Section 4.4 Khan Academy Calculus Videos Writing in Math/Discussion Research and prepare a report on the Fundamental Theorem of Calculus and the mathematicians Newton and Gottfried Leibniz.
understand when the antiderivative so represented is continuous and differentiable. Apply basic properties of definite integrals. Understand and use the Second Fundamental Theorem of Calculus. Domain: Calculate and Apply Integrals Cluster: Apply techniques of antidifferentiation Develop facility with finding antiderivatives that follow directly from derivatives of basic functions (power, exponential, logarithmic, and trigonometric). Use substitution of variables to calculate antiderivatives (including changing limits for definite integrals). Find specific antiderivatives using initial conditions. Objectives: Students will: Use pattern recognition to find an indefinite integral. Use change of variables to find an indefinite integral. Use the general power rule for integration to find an indefinite integral. Use a change of variables to evaluate a definite integral. Evaluate a definite integral involving an even or odd function. 4.5: Integration by Substitution Additional Resource(s) Larson Calculus Videos – Section 4.5 Khan Academy Calculus Videos Domain: Understanding Integrals Cluster: Demonstrate understanding of a Definite Integral 3. Use Riemann sums (left, right, and midpoint) and trapezoidal sums to approximate definite integrals of functions, represented graphically, numerically, and by tables of values. Objectives: Students will: Approximate a definite integral using the Trapezoidal Rule. 4.6: Numerical Integration Additional Resource(s) Larson Calculus Videos – Section 4.6 Khan Academy Calculus Videos