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Using the Definite Integral: Areas (4/20/09) You can use the definite integral to compute areas (and, later on in calculus, volumes, surface areas,arc lengths, etc.) which are bounded by curves we can describe. For example, what is the area (in square units) enclosed by f (x) = -x 2 + x + 2 and the x – axis? Find the area enclosed f (x) = e x, the line y = e, and the y – axis.

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Clicker Question 1 What is the area enclosed by f (x ) = 4 – x 2 and the x – axis? A. 16/3 B. 0 C. 32/3 D. 8 E. -16/3

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Integral Notation Notation: The symbol is traditionally used to denote integration of a function f (x). It is used in two ways: means the integral (or definite integral) of f over the interval [a, b]. Note that it is a number ! means the most general antiderivative (or “indefinite integral”) of f with respect to x. Note that it is a family of functions !

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Examples Find:

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Clicker Question 2 What is ? A. ½ x 2 + 1 B. ½ x 2 + ln|x| + C C. 1 – 1 /x 2 D. ½ x 2 – ln|x| + C E. 2x 2 + x + C

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Clicker Question 3 What is ? A. 1 B. 2 C. ½ sin(2x ) + /8 D. 2 / 2 E. 2 / 4

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Initial value problems The most general antiderivative of a function f (x) is a family of functions, all differing by a constant. If we are given a single data point about the antiderivative (an “initial condition”), then we can evaluate the constant term, getting a specific function.

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Examples & Assignment Find the antiderivative of f (x) = e 2x which satisfies that its value at 0 is 7. A car starts out 100 miles from NYC and travels away from it at a velocity of v (t) = 30 + 30 sin(t) miles per hour. Find a formula for the distance D of the car from NYC at t hours after the start. For Wednesday: Do sheet handed out (NOT hand-in).

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