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In the previous two sections, we looked at the definite integral and its relationship to the area under the curve of a function. Unfortunately, so far, the only tools we have available to calculate the value of a definite integral are geometric area formulas and limits of Riemann sums, and both approaches are extremely cumbersome. In this section we look at some more powerful and useful techniques for evaluating definite integrals.
These new techniques rely on the relationship between differentiation and integration. This relationship was discovered and explored by both Sir Isaac Newton and Gottfried Wilhelm Leibniz (among others) during the late 1600s and early 1700s, and it is codified in what we now call the Fundamental Theorem of Calculus , which has two parts that we examine in this section. Its very name indicates how central this theorem is to the entire development of calculus.
Isaac Newton ’s contributions to mathematics and physics changed the way we look at the world. The relationships he discovered, codified as Newton’s laws and the law of universal gravitation, are still taught as foundational material in physics today, and his calculus has spawned entire fields of mathematics. To learn more, read a brief biography of Newton with multimedia clips.
Before we get to this crucial theorem, however, let’s examine another important theorem, the Mean Value Theorem for Integrals, which is needed to prove the Fundamental Theorem of Calculus.
The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. The theorem guarantees that if $f\left(x\right)$ is continuous, a point c exists in an interval $\left[a,b\right]$ such that the value of the function at c is equal to the average value of $f\left(x\right)$ over $\left[a,b\right].$ We state this theorem mathematically with the help of the formula for the average value of a function that we presented at the end of the preceding section.
If $f\left(x\right)$ is continuous over an interval $\left[a,b\right],$ then there is at least one point $c\in \left[a,b\right]$ such that
This formula can also be stated as
Since $f\left(x\right)$ is continuous on $\left[a,b\right],$ by the extreme value theorem (see Maxima and Minima ), it assumes minimum and maximum values— m and M , respectively—on $\left[a,b\right].$ Then, for all x in $\left[a,b\right],$ we have $m\le f\left(x\right)\le M.$ Therefore, by the comparison theorem (see The Definite Integral ), we have
Dividing by $b-a$ gives us
Since $\frac{1}{b-a}{\displaystyle {\int}_{a}^{b}f}\left(x\right)dx$ is a number between m and M , and since $f\left(x\right)$ is continuous and assumes the values m and M over $\left[a,b\right],$ by the Intermediate Value Theorem (see Continuity ), there is a number c over $\left[a,b\right]$ such that
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