AP Calculus AB
Integration and Accumulation
Calculus AB
- ✓Apply the Fundamental Theorem of Calculus (both parts) to evaluate definite integrals and find derivatives of integrals.
- ✓Calculate the area of a region bounded by curves using definite integrals.
- ✓Determine the volume of a solid of revolution using the disk and washer methods.
- ✓Solve separable differential equations and interpret their solutions.
- ✓Use definite integrals to model and solve problems involving net change and total accumulation.
Key concepts
If f is continuous on [a, b], then the function g defined by g(x) = ∫[a to x] f(t) dt is continuous on [a, b] and differentiable on (a, b), and g'(x) = f(x). This part establishes the relationship between differentiation and integration as inverse processes. If the upper limit is a function of x, say u(x), then the Chain Rule must be applied.
If f is continuous on [a, b] and F is any antiderivative of f (meaning F'(x) = f(x)), then the definite integral of f from a to b can be evaluated by finding the difference of F at the upper and lower limits.
The definite integral of a rate of change function over an interval gives the net change (or total accumulation) of the quantity over that interval. For example, if v(t) is the velocity of an object, ∫[a to b] v(t) dt represents the net displacement. If |v(t)| is the speed, ∫[a to b] |v(t)| dt represents the total distance traveled.
To find the area of the region bounded by two continuous curves, integrate the difference between the 'top' and 'bottom' functions (or 'right' and 'left' functions) over the interval where they enclose a region. It is crucial to correctly identify which function is greater over the interval.
Used to find the volume of a solid of revolution when the region being revolved is flush against the axis of revolution. The cross-sections perpendicular to the axis of revolution are disks, and their volume is π * (radius)^2 * (thickness).
Used to find the volume of a solid of revolution when there is a gap between the region being revolved and the axis of revolution. The cross-sections perpendicular to the axis of revolution are washers (disks with a hole), and their volume is π * ((outer radius)^2 - (inner radius)^2) * (thickness).
A differential equation is separable if it can be written in the form dy/dx = g(x)h(y). To solve, separate the variables (move all y terms with dy and all x terms with dx) and then integrate both sides. Remember to include the constant of integration and use initial conditions to find particular solutions.
Key facts to remember
- 1The Fundamental Theorem of Calculus, Part 1: d/dx [∫[a to x] f(t) dt] = f(x), and d/dx [∫[a to u(x)] f(t) dt] = f(u(x)) * u'(x).
- 2The Fundamental Theorem of Calculus, Part 2: ∫[a to b] f(x) dx = F(b) - F(a), where F is any antiderivative of f.
- 3The definite integral ∫[a to b] f(x) dx represents the net accumulation of f(x) over the interval [a, b].
- 4Area between curves: ∫[a to b] (Top Curve - Bottom Curve) dx or ∫[c to d] (Right Curve - Left Curve) dy.
- 5Volume by Disk Method: V = π ∫[a to b] [R(x)]^2 dx (or R(y) for y-axis revolution).
- 6Volume by Washer Method: V = π ∫[a to b] ([R(x)]^2 - [r(x)]^2) dx (or R(y), r(y) for y-axis revolution).
- 7Separable differential equations are solved by isolating variables and integrating both sides.
- 8Always include the constant of integration '+ C' when finding indefinite integrals or solving differential equations.
Worked examples
Example 1
Evaluate d/dx [∫[1 to x^2] cos(t^3) dt] and ∫[0 to π/2] sin(x) dx.
Answer
d/dx [∫[1 to x^2] cos(t^3) dt] = 2x cos(x^6); ∫[0 to π/2] sin(x) dx = 1.
Example 2
Find the area of the region bounded by y = x^2 and y = 4. Then, find the volume of the solid generated by revolving this region about the line y = 4.
Answer
Area = 32/3; Volume = 512π/15.
Example 3
Find the particular solution to the differential equation dy/dx = (x+1)/y with the initial condition y(0) = 2.
Answer
y = √(x^2 + 2x + 4).
Remember to include the constant of integration 'C' immediately after integrating, and use the initial condition to determine its specific value for the particular solution. Pay attention to the sign of y based on the initial condition.
Common mistakes
- ✗Forgetting the Chain Rule when applying FTC1 if the upper limit of integration is a function of x (e.g., x^2).
- ✗Incorrectly identifying the 'top' and 'bottom' (or 'right' and 'left') functions for area calculations, leading to an incorrect setup or negative area.
- ✗Mixing up the inner and outer radii (r and R) in the washer method, or incorrectly determining the radius function relative to the axis of revolution.
- ✗Forgetting the 'π' in volume formulas or the '+ C' when solving indefinite integrals or differential equations.
- ✗Algebraic errors when solving for the constant of integration or simplifying complex expressions during evaluation.
Exam tips
- ★Clearly show all steps, especially when applying the Fundamental Theorem of Calculus or setting up integrals for area and volume problems. Partial credit is often awarded for correct setups.
- ★Draw a sketch of the region for area and volume problems. This helps visualize the setup, identify intersection points, and correctly determine radii and which function is 'top'/'bottom'.
- ★Pay close attention to the limits of integration and the variable of integration (dx or dy) to ensure consistency with the setup.
- ★For differential equations, remember to separate variables completely before integrating and to solve for the constant of integration using the given initial condition to find the particular solution.
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