Section 5.2 The Definite Integral

Last section we were concerned with finding the area under a curve We used rectangles in order to estimate that area We also found that the more rectangles we used, the better our estimate Ideally we would use an infinite amount of rectangles We will use our notion of limits and sigma notation to rewrite our left hand and right hand sums These sums are called Riemann Sums

Left-Hand Sum Notice the sum begins at t 0, the beginning of our interval When is this an overestimate? –When the function is always decreasing over the interval (called monotonic decreasing) When is this an underestimate? –When the function is always increasing over the interval (called monotonic increasing)

Right-Hand Sum Notice the sum ends at t n, the end of our interval When is this an overestimate? –When the function is always increasing over the interval (called monotonic increasing) When is this an underestimate? –When the function is always decreasing over the interval (called monotonic decreasing)

What if we wanted to find the area of between 0 and 3 using 5 rectangles? What if we wanted to be more accurate? What if we wanted to be exact? We call the exact area the definite integral

The Definite Integral We take the limit of Riemann sums as n goes to infinity. If f is continuous for a≤t≤b, the limits of the left-and right-hand sums exist and are equal. f is called the integrand, and a and b are called the limits of integration. –Let’s find the definite integral from our example

The DEFINITE INTEGRAL as AREA When a < b, and f(x) is positive for some x-values and negative for others: The sum of the areas above the x-axis (+) and those below the x-axis (-)

Examples