Calculus Animations,Graphics and Lecture Notes

Riemann Sums and the Fundamental Theorem of Calculus

About Kelly Liakos
Calculus 1- Limits and Derivatives
Calculus 1 - The Second Derivative
Calculus 1 and 3 Formula Sheets
Chain Rule
Computer Lab Assignments
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Line Integrals
Optimization and Related Rates
Optimization for Functions of 2 Variables
Parametric Equations 2-space
Parametric Equations 3-space
Partial Derivatives
Polar Coordinate System
Polar Coordinates- Derivatives and Integrals
Riemann Sums and the Fundamental Theorem of Calculus
2d order Diff EQS-Motion
2d Partial Derivatives
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Triple Integrals
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Visualizing Limits of Functions of 2 Variables
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Scratch Paper
2 Poems

Before viewing the following animations you might want to see the computer lab on Riemann Sums on the Computer Lab Page first for a complete discussion of Riemann Sums.

The first animation below show the left and right sums for f(x) = x2 on the interval [1,2]

The second animation shows the midpoint rule for the  same function and the same interval.

Note since   f(x) = x2 is an increasing function the right hand sum is an over estimate and the left hand sum is an under estimate.

The exact answer is 7/3 0r 2.333. Note the midpoint rule converges much faster than either the left hand or right hand rules.

Computer Lab Assignment page

Left and Right Sums

Midpoint Sum

The following 2 animations are for the function f(x) = e -x on [0,2].

The first animation are the left and right hand sums. this time since f(x) = e -x is decreasing the right hand sum is an under estimate and the left hand sum is is an over estimate.

The second animation is the midpoint rule.

Note the exact answer is .865.

Left and Right Sums

Midpoint Sum

Proof of the Fundamental Theorem of Calculus

The discussion of the trapezoid rule and Simpson's Rule-2 methods for approximating the definite integral are found on the Computer Lab Assignment Page.

Let R be the region bounded by the curves f(x) = ln(x2+1) and g(x) = cos(x)


a. Find the Area of R


B Suppose R is the base of a solid and the cross-sections taken perpendicular to the

base are isosceles right triangles-- Set up but do not evaluate the integral which calculates the Volume.

Solution to Area/Volume Question

The Smart Bunny-A very short story by Kurt Vonnegut Jr.

©2008-2010 Kelly Liakos

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