AP Calc Riemann Sums/Area and Volume of Curves

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

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)

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

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*

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:

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*

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:

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

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: