Riemann Sums The Definite Integral Computing Definite Integrals on a Calculator Integrability … and why The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus. Slide 5- 3
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.
Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable)
We have the notation for integration, but we still need to learn how to evaluate the integral.
time velocity After 4 seconds, the object has gone 12 feet. 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.
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.
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!