1. § 6.3
Definite Integrals and the Fundamental Theorem

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

3. 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

4. Calculating Definite Integrals
EXAMPLE
Calculate the following integral.
SOLUTION
The figure shows the graph of the function f (x) = x 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

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

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

7. 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 Therefore, from geometry we find that
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #35

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

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

10. 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

11. 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

12. 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 degrees Fahrenheit.
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #40

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