AP® Calculus AB & BC All Access Book + Online by Stu Schwartz

Author:Stu Schwartz
Language: eng
Format: epub
Publisher: Research & Education Association
Published: 2016-01-15T00:00:00+00:00

When you see a definite integral, you must always think of it as a representation of an area. In geometry you learned that area is always a positive number. However, it is possible for a definite integral to be negative. Here’s how:

When a < b, we determine the area under the curve from left to right. Our dx will be a positive number.

When b < a, we determine the area under the curve from right to left. Our dx will be a negative number.

This can be summarized below.

Furthermore, there are several rules that make absolute sense when you think of definite integrals as areas. If f is continuous on the closed interval [a, b], it follows that:

1.— If you start at a and end at a, there is no area.

2. Suppose f(x) is positive. will be positive so will be negative which will be . You can always change the order of the limits in a definite integral if you apply a negative sign to the integral sign.

3.— Add the area from a to b to the area from b to c and you get the area from a to c. It makes no difference if the function is above or below the x–axis.

4. where k is a constant — The area under k. f(x) is k times the area under f(x).

5.— The area under f(x) plus (or minus) g(x) is the area under f(x)=g(x) or f(x)–g(x).

EXAMPLE 13:For the following problems suppose f(x) and g(x) are given by the following graphs, made up of lines and a semi-circle. Evaluate each part.

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