14.6: Definite Integrals Notes

14.6: Definite Integrals

Definition of Definite Integral

The definite integral of a function f(x) from a to b, denoted as \int_{a}^{b} f(x) dx, represents the area under the curve y = f(x) between the lines x = a and x = b. Here,

• a and b are called the "bounds" or "limits of integration."

• a is the "lower bound" or "lower limit."

• b is the "upper bound" or "upper limit."

\int_{a}^{b} f(x) dx is called a "definite integral."

Approximating Definite Integrals

Definite integrals are calculated by approximating the area under the curve using rectangles.

• Lower Sum: An underestimate of the area.

• Upper Sum: An overestimate of the area.

Approximating Sums

These approximating sums have the form:

f(x1) \Delta x + f(x2) \Delta x + \ldots + f(xn) \Delta x = \sum{i=1}^{n} f(x_i) \Delta x

The x_i are called "sample points"; they are the points where the height of the rectangle is measured.

• Right Endpoints: x_i is the right edge of the rectangle.

• Left Endpoints: x_i is the left edge of the rectangle.

Example: Approximating \int_{0}^{1} 2x dx

Consider the function f(x) = 2x with a = 0 and b = 1. Let's find \int{0}^{1} f(x) dx = \int{0}^{1} 2x dx.

Dividing the interval into n subintervals gives:

x0 = 0, x1 = \frac{1}{n}, x2 = \frac{2}{n}, \ldots, xn = \frac{n}{n} = 1

The width of the rectangles is \Delta x = \frac{1}{n}.

The heights of the rectangles are:

f(x1) = 2 \cdot \frac{1}{n} = \frac{2}{n}, f(x2) = 2 \cdot \frac{2}{n} = \frac{4}{n}, \ldots, f(x_n) = 2 \cdot \frac{n}{n} = 2

So the sum using right endpoints is:

Sn = \sum{i=1}^{n} f(x_i) \Delta x = \frac{2}{n} \cdot \frac{1}{n} + \frac{4}{n} \cdot \frac{1}{n} + \ldots + \frac{2n}{n} \cdot \frac{1}{n} = \frac{2}{n^2} (1 + 2 + \ldots + n)

Using the arithmetic progression formula:

1 + 2 + \ldots + n = \frac{n(n+1)}{2}

We have:

S_n = \frac{2}{n^2} \cdot \frac{n(n+1)}{2} = \frac{n(n+1)}{n^2} = \frac{n^2 + n}{n^2} = \frac{n+1}{n}

Thus,

\int{0}^{1} 2x dx = \lim{n \to \infty} Sn = \lim{n \to \infty} \frac{n+1}{n} = \lim_{n \to \infty} \frac{n}{n} = 1

Key Facts

• \int_{a}^{b} f(x) dx is the limit of approximating sums.

• Larger values of n give better approximations.

• Changing the sample points can affect whether the sum is an upper or lower sum.

Examples

Example 1: Left and Right Sums

Suppose f(1) = 3, f(2) = 4, f(3) = 3, and f(4) = 2. Find left and right sums approximating \int_{1}^{4} f(x) dx.

• \Delta x = 1

• Left Sum: f(1) \cdot \Delta x + f(2) \cdot \Delta x + f(3) \cdot \Delta x = 3 \cdot 1 + 4 \cdot 1 + 3 \cdot 1 = 10

• Right Sum: f(2) \cdot \Delta x + f(3) \cdot \Delta x + f(4) \cdot \Delta x = 4 \cdot 1 + 3 \cdot 1 + 2 \cdot 1 = 9

Example 2: Another Left and Right Sum

Suppose g(1) = 0, g(\frac{3}{2}) = 1, g(2) = 2, g(\frac{5}{2}) = 4, and g(3) = 8. Find left and right sums approximating \int_{1}^{3} g(x) dx.

• \Delta x = \frac{1}{2}

• Left Sum: g(1) \cdot \Delta x + g(\frac{3}{2}) \cdot \Delta x + g(2) \cdot \Delta x + g(\frac{5}{2}) \cdot \Delta x = 0 \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{2} + 4 \cdot \frac{1}{2} = \frac{7}{2}

• Right Sum: g(\frac{3}{2}) \cdot \Delta x + g(2) \cdot \Delta x + g(\frac{5}{2}) \cdot \Delta x + g(3) \cdot \Delta x = 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{2} + 4 \cdot \frac{1}{2} + 8 \cdot \frac{1}{2} = \frac{15}{2}

Example 3: Car's Speedometer

A car's speedometer reads 20 mph at noon, 30 mph at 12:30 pm, 20 mph at 1 pm, and 40 mph at 1:30 pm. Estimate the distance traveled using left and right sums.

• \Delta t = \frac{1}{2} (half an hour)

• Left Sum: 20 \cdot \frac{1}{2} + 30 \cdot \frac{1}{2} + 20 \cdot \frac{1}{2} = 10 + 15 + 10 = 35

• Right Sum: 30 \cdot \frac{1}{2} + 20 \cdot \frac{1}{2} + 40 \cdot \frac{1}{2} = 15 + 10 + 20 = 45

Note: 30 \cdot \frac{1}{2} means "half an hour at 30 mph."

Example 4: Approximating \int_{2}^{4} x^2 dx

Approximate \int_{2}^{4} x^2 dx using left and right sums with n = 3.

First, divide [2, 4] into 3 subintervals:

x0 = 2, x1 = \frac{8}{3}, x2 = \frac{10}{3}, x3 = 4 and \Delta x = \frac{2}{3}

Next, evaluate f(x) = x^2 at each point:

f(x0) = 4, f(x1) = \frac{64}{9}, f(x2) = \frac{100}{9}, f(x3) = 16

• Left Sum:

f(x0) \cdot \Delta x + f(x1) \cdot \Delta x + f(x_2) \cdot \Delta x = 4 \cdot \frac{2}{3} + \frac{64}{9} \cdot \frac{2}{3} + \frac{100}{9} \cdot \frac{2}{3} = \frac{8}{3} + \frac{128}{27} + \frac{200}{27} = \frac{72 + 128 + 200}{27} = \frac{400}{27}

• Right Sum:

f(x1) \cdot \Delta x + f(x2) \cdot \Delta x + f(x_3) \cdot \Delta x = \frac{64}{9} \cdot \frac{2}{3} + \frac{100}{9} \cdot \frac{2}{3} + 16 \cdot \frac{2}{3} = \frac{128}{27} + \frac{200}{27} + \frac{32}{3} = \frac{128 + 200 + 288}{27} = \frac{616}{27}