Page 31 - Calc for the Clueless

Example 24—

If, , find y.

Let u = x + 11. du/dx = 4x .
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3

Example 25—

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If v = 6t + 4t + 3 and s = 40 when t = 1, find s. s = distance, v = velocity, t = time, v = ds/dt.

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Use t = 1 when s = 40 to get 40 = 2(1) + 2(1) + 3(1) + C. C=33.

The development of the area as motivation for the definite integral is detailed in most calculus books. We will
sketch the development.

A. Given the region y = f(x), x = a, x = b, x axis.

B. Divide the interval [a,b] into n intervals, ∆x 1, ∆x 2, ∆x 3,... ∆x n. ∆x i represents an arbitrary interval.

C. Let w i be any point in the interval ∆x i.

D. ∆x 1 represents the width of the first approximating rectangle. f(w 1) represents the height of the first rectangle.
∆x 1 f(w 1) represents the area of the first approximating rectangle.

E. Do this for all the rectangles. We get