Chegg calculus

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Chegg calculus

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Chegg calculus

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 s and early s, 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. To learn more, read a brief biography of Newton with multimedia clips. The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at some point in that interval. The theorem guarantees that if f x f x is continuous, a point c exists in an interval [ a , b ] [ a , b ] such that the value of the function at c is equal to the average value of f x f x over [ a , b ].

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