Area & Riemann Sums Chapter 5.1

Distance Traveled Suppose that you drive on a straight highway at a constant velocity of 72 mph for 1.5 hours, and then at 60 mph for 2 hours. How far are you (what is your displacement) from your starting point? Your displacement can be viewed as the area between the velocity function and the t-axis. In this example, your total displacement is 228 miles. 80 40 velocity Area = 108 Area = 120 1 2 3 4 time

More Realistic – Distance Traveled Velocity is rarely constant for an extended period of time, as moving objects need to speed up or slow down. How might we find the displacement of this moving object over the interval 0 < t < 4 ?

The Area Problem Video

Area Under a Curve Assume that f (x) ≥ 0 and f is continuous on the interval [a, b]. To approximate the area under the curve, we start by dividing the interval [a, b] into n equal pieces. This is called a regular partition of [a, b]. The width of each subinterval in the partition is , which we denote by Δx, meaning a small change in x).

Riemann Sum On each subinterval (for i = 1, 2, . . . , n), construct a rectangle of height f (xi*), where xi* is an x-coordinate in the ith subinterval. For example, xi* can be the left endpoint of the ith subinterval, the right endpoint, or the midpoint. The area under the curve, A, is roughly the same as the sum of the areas of the rectangles. The more rectangles, the better the approximation. Sums like this are called Riemann Sums.

Left-endpoint Riemann Sum 2 3 5 6 5 3 2

Right-endpoint Riemann Sum 3 5 6 5 3 2 3

Example Approximate the area under the graph of f (x) = 2x over the interval [0, 4] using a right-endpoint Riemann sum with n = 8 rectangles. (4, 8)

Sigma notation A way to abbreviate a sum of similar terms is to use sigma notation. Similarly, ending number typical term, a function of i starting number

Sums of the form We divide the interval [a, b] into n equal parts. The width of each part is . Within each subinterval, we choose an x-coordinate. The x-coordinate we select in the ith subinterval is denoted xi* . We use the chosen x-coordinate to find the height of the rectangle over that subinterval: f (xi* ). The area of the ith rectangle is Ai = f (xi* ) Δx. To approximate the total area, we add all of the rectangles, from the first to the nth.

Example Compute the sum for f (x) = x2 + 5x, with xi* = 0.2, 0.4, 0.6, 0.8, and 1.0, Δx = .2, n = 5 We want to add [.22 + 5(.2)](.2) + [.42 +5(.4)](.2) + … + [12 + 5(1)](.2) =3.44

Background Material In order to compute the limit of a Riemann Sum, we need to know a few facts about sums. Sum of constant terms: Addition/subtraction: Constant multiple:

Special sums Sum of the first n positive integers 2. Sum of the first n squares

Using Shortcuts Use the special sums and properties of sums to find 1. 2. A. 9190 B. 8350 C. 8198 D. 1177

Definite Integrals 5.2

Finding the Actual Area To approximate the area under a curve, we computed sums like . The actual area under the curve is the limit of such a sum as Δx  0 or as n ∞. If f is a continuous, non-negative function over the interval [a, b], then the area under the graph of f(x) over [a, b] is .

Find the Actual Area Find the area under the graph of f (x) = 2x over the interval [0, 4] by finding the limit of a right-endpoint Riemann sum as n  ∞. (4, 8)

Another Example Use the limit of a Riemann sum to compute the exact area under y = x2 + 2 over the interval [0, 1].

The Definite Integral Definition: For any function f defined on an interval [a, b], the definite integral of f from a to b is whenever the limit exists and is the same for every choice of evaluation points, x1*, x2*, . . . , xn*. When the limit exists, we say that f is integrable over the interval [a, b].

THE DEFINITE INTEGRAL The elongated “S” is the integral sign. The lower and upper limits of integration, a and b, respectively, indicate the endpoints of the interval over which you are integrating. The dx in the integral corresponds to the increment Δx in the Riemann sum and also indicates the variable of integration. f (x) is called the integrand.

Definite Integrals and Area When f (x) > 0 and is continuous over the interval [a, b], the area under the graph of f (x) over the interval is Now, we have defined the definite integral to be Therefore, for a non-negative function, the definite integral equals the area under the curve.

Computing a Definite Integral We have options: If the integral represents an area that we know from a geometry formula, we can calculate it directly. If the function is simple enough, we can compute the limit of the Riemann sum. We can approximate the integral with a Riemann sum with a large n. We can use the graphing calculator to approximate the integral.

Definite Integrals - Examples 1. 2. 3.

What if f (x) is negative over [a, b]? If f (x) negative over [a, b], then will also be negative. For the function shown here, the area between the curve and the x-axis is 4/3. Area = 4/3

Voting Question Use a formula from geometry to evaluate A. -2 B. -4 C. 2 D. 4 2 -2

What if f is both positive and negative over an interval? The definite integral gives the Net Area Net Area: The sum of the areas of the parts of the region that lie above the x-axis minus the sum of the areas of the parts that lie below the x-axis

What if f (x) is both positive and negative over an interval? In this example, the net area is 0. But, the total area is We can obtain this using two integrals:

Voting Question Earlier, we found out that What is the value of A. 0 B. 2 C. -2 D. 4

Properties of Definite Integrals

Given that and , find: 1. A. 4 B. 8 C. -12 D. -3 2. A. 4 B. 10 C. 24 D Given that and , find: 1. A. 4 B. 8 C. -12 D. -3 2. A. 4 B. 10 C. 24 D. 40 3. A. -2 B. 2 C. 4 D. -12