Week 12 Calculus I Outline of Notes

by S. Gramlich

(updated 12/11/06)

I) Chapter 5: Finding Areas Under the Curve with curved sides

5.1 & 5.2: Using Riemann Sums on continuous f(x) on interval [a,b] with n subintervals

A = lim _{(n->∞) i=1}∑ⁿ f(x) Δx = Δx [f(x₁) + f(x₂) + ... + f(x_n)]

where A=Area, f(x)= height/length, Δx = (b-a)/n =base/width

p. 375, can sum rectangles where the estimate of A uses:

p. 370, Fig 4b, R_n=Right Endpoints coincide with curve (use x_i), start @ a + Δx

p. 370, Fig 5, L_n=Left Endpoints coincide with curve (use x_i-1), start @ a = x₀

p. 374, Fig 13, Sample Points (in middle somewhere) that coincide with curve (x_i*)

p. 386, Fig 11, M_n=Midpoints coincide with curve (use ˜_i), start @ (a+ (a+Δx))/2

where ˜_i = (x_i-1 + x_i)/2 = midpoint of [x_i-1, x_i]

Note: The more rectangles, the better the approximation of A

5.2, p. 383, Properties of Summation (7-10)

p. 370, 5.1 Example 1 (using Right, Left, Mid)

II) 5.2 Definition of Definate Integral

_a∫^b f(x) dx = lim _{(n->∞) i=1}∑ⁿ f(x_i*) Δx

where ∫= "integral," a= "upper limit", b= "lower limit", f(x)= "integrand"

5.2, p. 391, #20

pp. 387-389, Properties of Definate Integral

5.2, p. 392, #48

III) 5.3 THE FUNDAMENTAL THEREOM OF CALCULUS (FTC)

Part 1 (FTC1)

if f continuous on [a,b] then

g(x) = _a∫^x f(t) dt

so g'(x) = lim _(h->0) [g(x+h) - g(x)]/h = f(x),

f is the derivative of g or g is the antiderivative of f

* make sure upper limit is x, if not change it using a substitution

5.3, p. 398, Example 4

Part 2 (FTC2)

if f continuous on [a,b] then

_a∫^b f(x) dx = F(b) - F(a) = F(x)]_a^b

where F is any antiderivative of f

5.3, p. 400, Example 6

* Beware of discontinuities on the interval and negative answers for Area

5.3, p. 401, Example 9