1. AP Calc Riemann Sums/Area and Volume of Curves
By Ashwin, Jeffrey, Jake

2. Card #42:Accumulation (Riemann Sums)
Right/Left Riemann Sum:
Step 1:Draw rectangles whose upper corner aligns with the curve f(x)
Right sum-right corner on curve
Left sum-left corner on curve
Step 2: Find area of rectangles and add together
*area:bh
Trapezoidal Riemann Sum:
Step 1: Connect points by a straight line from point to point, draw trapezoids
Step 2: Find area of trapezoids and add together
*area:½(b(h1+h2)

3. Pictures/Justification
If f(x) is concave down, the trapezoidal sum is an underestimate
If f(x) is concave up, the trapezoidal sum is an overestimate
If f(x) is increasing:
Right Riemann sum is an overestimate, Left Riemann sum is an underestimate
If f(x) is decreasing:
Right Riemann sum is an underestimate, Left Riemann sum is an overestimate

4. Card #43: Area Between Curves
Step 1:Determine which functions are top/bottom or right/left on the graph
*If the functions are left/right of each other, write equations in terms of y
Step 2: Set equations equal to find intersections (or use given bounds)
Step 3: Plug into following formula: *f(x) is top or right and g(x) is left or bottom*

5. Card #44: Volume with Disk and Washer Method
Volume with Disk Method:
Step 1: Determine whether rotation occurs at a vertical or horizontal line.
*If rotating on a vertical line, write equations in terms of y
Step 2: Determine bounds by finding intersection with line of rotation (or use given bounds)
Step 3: Plug into following formula:

6. Card #44 - Continued Volume with Washer Method:
Step 1: Determine whether rotation occurs at a horizontal or vertical line
*If rotating on a vertical line, write equations in terms of y
Step 2: Determine bounds by finding intersection with axis (or use given information)
Step 3: Determine which functions are closer and further away from line of rotation.
Step 4: Plug into following formula: *F(x) is the function farthest away from line of rotation and G(X) is closest to line of rotation*

7. Card #45 Volume of Cross Section
Step 1: Determine if problem is perpendicular to x or y axis
Step 2: Determine the bounds of the integral by finding intersection with axis (or use given information)
Step 3: Find A(x) (on the next page)
Step 4: Plug into following formula:

8. Finding A(x)
*A(x) depends on the shape of the cross section that is used
*The functions on the graph represent “s”, “b” or diameter in a circle. If you have 2 functions, use (top-bottom) or (right-left) and substitute into s,b, or d
Common Formulas:
Square:s^2, Rectangle:(s)(h) *h is given
Triangle:0.5(b)(h) *h is given
Equilateral Triangle:√3/4(s)^2
Circle:π(r)^2 *must divide functions by 2 to substitute for r

9. Card #46: Arc Length Rectangular
Step1: Determine the bounds of the integral by finding intersections (or use given information)
*if y values are given, write equation in terms of y
Step 2: Plug into following formula: