Sect. 4.1 Antiderivatives Sect. 4.2 Area Sect. 4.3 Riemann Sums/Definite Integrals Sect. 4.4 FTC and Average Value Sect. 4.5 Integration by Substitution Sect. 4.6 Numerical Integration and Trapezoidal Rule Particle Motion CHAPTER FOUR: INTEGRATION

2 Section 4.3 Riemann Sums & the Definite Integral Goals Understand the definition of Riemann Sum & their relationship to integrals. Evaluate a definite integral using limits. Evaluate a definite integral using properties.

Area under a curve The other day we found the area under a curve by dividing that area up into rectangles and taking their sum (Riemann Sum). This section goes one step further and examines functions that are continuous, but may not lie above the x-axis.

When functions are non-negative, the Riemann sums estimates the areas under the curves, above the x-axis, over some interval [a, b]. When functions are negative, however, the Riemann sums estimate the negative (or opposite) values of those areas. In other words, Riemann sums have direction and CAN take on negative values.

Finer partitions of [a, b] create more rectangles with shorter bases. This is read “the integral from a to b of f of x dx,” or “the integral from a to b of f of x with respect to x.” ||∆|| denotes the width of the largest rectangle.

Formal Definition from p. 267 of your textbook

Area under the curve the definite integral of f from a to b The Definite Integral – Alternate Definition n denotes the number of rectangles.

Negative Values If f (x) is positive for some values of x on [a, b] and negative for others, then the definite integral symbol represents the cumulative sum of the signed areas between the graph of f (x) and the x axis, where areas above are positive and areas below negative. y = f (x) a bA B

Examples Calculate the definite integrals by referring to the figure with the indicated areas. Area A = 3.5 Area B = 12 y = f (x) a bA B c

For the next few slides, evaluate the definite integrals using geometric area formulas.

Top half only! circle centered at the origin with radius 2

You Try… Set up a definite integral for finding the area of the shaded region. Then use geometry to find the area. rectangle triangle

You Try… Use geometry to compute. Does this represent the area of the given region? Explain. Area = 2 Area =4

Estimate using a right hand Riemann sum and 6 partitions. (Estimate the area of the given region.) Example using Riemann Sum The right endpoints are: x 1 = 0.5, x 2 = 1.0, x 3 = 1.5, x 4 = 2.0, x 5 = 2.5, x 6 = 3.0

So, the Riemann Sum is: Notice that f is not a positive function. So, the Riemann Sum does NOT represent a sum of areas of rectangles.

You Try… Estimate using a left hand Riemann sum and 4 partitions.

Properties of Definite Integrals

ab c y x Additive Interval Property

Ex. If it is known that find:

More Properties of Integrals

EX.

You Try… Assume we know that A) B) Solve

You Try… (continued) C) D) E)

“Preservation of Inequality” Theorem If f is integrable and f(x) > 0 on [a, b] then If f and g are integrable and non-negative on [a, b] and f(x) < g(x) for every x in [a, b], then y = f(x ) ab y x

D C I Relationship between Differentiability, Continuity, and Integrability D – differentiable functions, strongest condition … all Diff ’ble functions are continuous and integrable. C – continuous functions, all cont functions are integrable, but not all are diff ’ble. I – integrable functions, weakest condition … it is possible they are not con‘ t, and not diff ‘ble. sinx, e x, x 2

Closure Explain the difference between the integral of the function pictured below on the interval [a, b] and the area of the same region. ab f A1A1 A2A2 A3A3 c d