5.2 Definite Integrals and Areas. The student will learn about: the definition of the definite integral, the fundamental theorem of calculus, and some applications.

Introduction We begin this section by calculating areas under curves, leading to a definition of the definite integral of a function. The Fundamental Theorem of Integral Calculus then provides an easier way to calculate definite integrals using indefinite integrals. Finally, we will illustrate the wide variety of applications of definite integrals.

Definite Integral as a Limit of a Sum. The Definite Integral may be viewed as the area between the function and the x-axis.

APPROXIMATING AREA BY RECTANGLES We may approximate the area under a curve Inscribing rectangles under it. Use rectangles with equal bases and with heights equal to the height of the curve at the left-hand edge of the rectangles.

Area Under a Curve The following table gives the “rectangular approximation” for the area under the curve y = x 2 for 1 ≤ x ≤ 2, with a larger numbers of rectangles. The calculations were done on a graphing calculator, rounding answers to three decimal places. # Rectangles Sum of Areas 4 2.71875 8 2.523438 16 2.427734 32 2.380371 64 2.356812 128 2.345062 256 2.339195 512 2.336264 1024 2.334798 2048 2.334066

Those Responsible. Isaac Newton 1642 -1727 Gottfried Leibniz 1646 - 1716

Example 1 5 · 3 – 5 · 1 = 15 – 5 = 10 Make a drawing to confirm your answer. 0  x  4 - 1  y  6

Example 2 4 Make a drawing to confirm your answer. 0  x  4 Nice red box?

Fundamental Theorem of Calculus If f is a continuous function on the closed interval [a, b] and F is any antiderivative of f, then

Evaluating Definite Integrals By the fundamental theorem we can evaluate Easily and exactly. We simply calculate

Definite Integral Properties

Example 3 9 - 0 = 9 0  x  4 - 2  y  10 Do you see the red box?

Example 4 3.6268604 There is that red box again?

Examples 5 This is a combination of the previous two problems = 9 + (e 6)/2 – 1/3 – (e2)/2 = 206.68654 What red box?

Numerical Integration on a Graphing Calculator 0  x  3 - 1  y  3 -1  x  6 - 0.2  y  0.5

Application From past records a management services determined that the rate of increase in maintenance cost for an apartment building (in dollars per year) is given by M ’ (x) = 90x 2 + 5,000 where M is the total accumulated cost of maintenance for x years. Write a definite integral that will give the total maintenance cost through the seventh year. Evaluate the integral. 30 x 3 + 5,000x = 10,290 + 35,000 – 0 – 0 = $45,290

Total Cost of a Succession of Units The following diagrams illustrate this idea. In each case, the curve represents a rate, and the area under the curve, given by the definite integral, gives the total accumulation at that rate.

FINDING TOTAL PRODUCTIVITY FROM A RATE A technician can test computer chips at the rate of –3x2 + 18x + 15 chips per hour (for 0 ≤ x ≤ 6), where x is the number of hours after 9:00 a.m. How many chips can be tested between 10:00 a.m. and 1:00 p.m.?

Solution - N (t) = –3t 2 + 18t + 15 The total work accomplished is the integral of this rate from t = 1 (10 a.m.) to t = 4 (1 p.m.): Use your calculator = ( - 64 + 144 + 60) – (-1 + 9 + 15) = 117 That is, between 10 a.m. and 1 p.m., 117 chips can be tested.

Summary. We can evaluate a definite integral by the fundamental theorem of calculus:

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