4.3: Definite Integrals Learning Goals Express the area under a curve as a definite integral and as limit of Riemann sums Compute the exact area under a curve using technology. ©2007 Roy L. Gover (

Riemann Sum If f is a continuous function, then the left Riemann sum with n equal subdivisions for f over the interval [a, b] is defined to be

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width of a rectangle is called a subinterval. The entire interval is called the partition. Subintervals do not all have to be the same size.

subinterval partition If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better. if P is a partition of the interval

is called the definite integral of over. If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by:

Leibniz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

The Definite Integral If f is a continuous function, the definite integral of f from a to b is defined to be The function f is called the integrand, the numbers a and b are called the limits of integration, and the variable x is called the variable of integration.

Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.

We have the notation for integration, but we still need to learn how to evaluate the integral.

The Definite Integral As a Total If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

The Definite Integral As a Total Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

If the velocity varies: Distance: ( C=0 since s=0 at t=0 ) After 4 seconds: The distance is still equal to the area under the curve! Notice that the area is a trapezoid.

Area Under a Graph abab Idea: To find the exact area under the graph of a function. Method: Use an infinite number of rectangles of equal width and compute their area with a limit. Width: (n rect.)

Approximating Area Approximate the area under the graph of using n = 4.

Area Under a Graph a b f continuous, nonnegative on [a, b]. The area is

What if: We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example. It seems reasonable that the distance will equal the area under the curve.

The area under the curve We can use anti-derivatives to find the area under a curve!

Area

The Definite Integral

k th Rectangle Let’s unpack those last two slides A representative rectangle

Definition A Riemann Sum is the sum of the area of all the rectangles height width

a b Let be the width of the largest rectangle. As approaches 0, the width of all rectangles approach 0.

As the width of all rectangles approach 0, the number of rectangles, n, approaches infinity a b

a b As the width of the rectangles approach 0, the portion of the rectangles above or below the curve (error) approaches 0.

As widths of the rectangles become more narrow, the right end point, left end point & midpoint merge to the same point. a b

Definition If f is defined on the closed interval [ a,b ] and the limit exists, then f is integrable on [ a, b ] and the limit is denoted by:

Definition The Definite Integral from a to b: Lower limit if integration Upper limit of integration Integrand Variable of integration

Properties of the Definite Integral 1: 2: 3: 4: 5: 6:

7:

Area from x=0 to x=1 Example: Find the area under the curve from x = 1 to x = 2. Area from x=0 to x=2 Area under the curve from x = 1 to x = 2.