MthT 430 Notes Chapter 5b Geometric Limits

Geometric Limits

Let A or A(H) denote the cross sectional area at height H.

For Beaker X,

∆H ∆V = 1 A , lim ∆V → 0  ∆H ∆V = 1 A .

For the Ink Bottle,

∆H ∆V ≈ 1 A(H) , lim ∆V → 0  ∆H ∆V = 1 A(H) .

Notice that for a circular flask A = π(radius)².

Fundamental Theorem of Calculus

Let y = f(x) be a nonnegative continuous function and let A(x) denote area of the region

(t,y): ⎧ ⎪ ⎨ ⎪ ⎩ a ≤ t ≤ x, a ≤ y ≤ f(x).

This is the region with first coordinate between a and x , second coordinate between the x-axis and the graph of f .

Now the change in A(x) as x changes from x to x + ∆x,

∆A = A(x + ∆x) − A(x),

is the area of the slender region and is

∆A ≈ f(x) ·∆x;

certainly ∆A is between f(x) ·∆x and f(x + ∆x) ·∆ x. Thus if f is continuous at x,

lim ∆x → 0  ∆A ∆x = f(x).

Identifying area with the integral, this is one form of the Fundamental Theorem of Calculus (FTC):

Theorem. If f is continuous on [a,b], then for a < x < b,

d dx ⌠ ⌡ x a  f(t)  dt = f(x).

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On 22 Aug 2014, 04:37.