5.2 Definite Integrals

1. 5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon , Siena College

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

3. 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. subinterval partition if P is a partition of the interval

4. 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:

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

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

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

8. velocity time In section 5.1, we considered 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. After 4 seconds, the object has gone 12 feet.

9. 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.

10. 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.

11. The area under the curve

12. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 1. Constant multiples can be moved outside. 3. There are rules for working with integrals, the most important of which are:

13. 4. Integrals can be added and subtracted. Reversing the limits changes the sign. 1. If the upper and lower limits are equal, then the integral is zero. 2. Constant multiples can be moved outside. 3.

14. 5. Intervals can be added (or subtracted.) 4. Integrals can be added and subtracted.