Write as the limit of a Riemann Sum.
SOLUTION: The endpoints indicate that a = 1 and b = 4, so and
. Therefore, we can write the definite integral as
.
D. THE FUNDAMENTAL THEOREM AGAIN
Area
If f(x) is nonnegative on [a,b], we see (Figure N6–1) that f(xk) Δxk can be
regarded as the area of a typical approximating rectangle. As the number of
rectangles increases, or, equivalently, as the width Δx of the rectangles
approaches zero, the rectangles become an increasingly better fit to the curve.
The sum of their areas gets closer and closer to the exact area under the curve.
Finally, the area bounded by the x-axis, the curve, and the vertical lines x = a and
x = b is given exactly by
Figure N6–1