§ 6.3 Definite Integrals and the Fundamental Theorem

Section Outline The Definite Integral Calculating Definite Integrals The Fundamental Theorem of Calculus Area Under a Curve as an Antiderivative

The Definite Integral Δx = (b – a)/n, x1, x2, …., xn are selected points from a partition [a, b]. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #31

Calculating Definite Integrals EXAMPLE Calculate the following integral. SOLUTION The figure shows the graph of the function f (x) = x + 0.5. Since f (x) is nonnegative for 0 ≤ x ≤ 1, the definite integral of f (x) equals the area of the shaded region in the figure below. 1 1 0.5 Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #32

Calculating Definite Integrals CONTINUED The region consists of a rectangle and a triangle. By geometry, Thus the area under the graph is 0.5 + 0.5 = 1, and hence Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #33

The Definite Integral Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #34

Calculating Definite Integrals EXAMPLE Calculate the following integral. SOLUTION The figure shows the graph of the function f (x) = x on the interval -1 ≤ x ≤ 1. The area of the triangle above the x-axis is 0.5 and the area of the triangle below the x-axis is 0.5. Therefore, from geometry we find that Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #35

Calculating Definite Integrals CONTINUED Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #36

The Fundamental Theorem of Calculus Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #37

The Fundamental Theorem of Calculus EXAMPLE Use the Fundamental Theorem of Calculus to calculate the following integral. SOLUTION An antiderivative of 3x1/3 – 1 – e0.5x is . Therefore, by the fundamental theorem, Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #38

The Fundamental Theorem of Calculus EXAMPLE (Heat Diffusion) Some food is placed in a freezer. After t hours the temperature of the food is dropping at the rate of r(t) degrees Fahrenheit per hour, where (a) Compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2. (b) What does the area in part (a) represent? SOLUTION (a) To compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2, we evaluate the following. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #39

The Fundamental Theorem of Calculus CONTINUED (b) Since the area under a graph can represent the amount of change in a quantity, the area in part (a) represents the amount of change in the temperature between hour t = 0 and hour t = 2. That change is 24.533 degrees Fahrenheit. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #40

Area Under a Curve as an Antiderivative Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #41