Riemann Sums and the Fundamental Theorem of Calculus

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 (ftc)

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

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