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The First Fundamental Theorem of Calculus

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First Fundamental Theorem Take the Antiderivative. Evaluate the Antiderivative at the Upper Bound. Evaluate the Antiderivative at the Lower Bound. Subtract the Lower Bound Value from the Upper Bound Value.

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The Definite Integral If f is a continuous function, the definite integral of f from a to b is defined to be The function f is called the integrand, the numbers a and b are called the limits of integration, and the variable x is called the variable of integration.

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The Definite Integral is read “the integral, from a to b of f(x)dx.” Also note that the variable x is a “dummy variable.”

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The Definite Integral As a Total If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

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Net Change This can be rewritten as follows The quantity F(b) – F(a) is the net change of the function F over the interval [a,b]. The derivative F’(x) is the rate of change of the function F. By the Fundamental Theorem of Calculus we have, for an indefinite integral function F of f: Definition

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So now we can do all kinds of summing problems before we even mention an antiderivative. Historically, that’s what scientists had to do before calculus. Here’s why it mattered to them:

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The Definite Integral As a Total Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

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The calculus pioneers knew that the area would still yield distance, but what was the connection to tangent lines? And was there an easy way to find these irregularly- shaped areas? The Definite Integral

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Area Under a Graph a b Idea: To find the exact area under the graph of a function. Method: Use an infinite number of rectangles of equal width and compute their area with a limit. Width: (n rect.)

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Geometric Interpretation (All Functions) Area of R 1 – Area of R 2 + Area of R 3 a b R1R1 R2R2 R3R3

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Area Using Geometry Ex. Use geometry to compute the integral Area = 2 Area =4

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Computing Area Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x 3 is nonnegative on [0, 2]. AntiderivativeFund. Thm. of Calculus

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Evaluate:

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Evaluate Definite Integrals with your calculator. Enter the function in y =. GRAPH & CALC (2 nd TRACE) Option 7 or MATH Option 9 fnInt(y1,x, __, __) Enter Lower Bound Enter Upper Bound

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AREA under f (x) down to the x-axis from x = a to x = b Use your graphing calculator to graph the integrand and determine whether the integral is… positive, negative, or zero.

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