AP Calculus AB HW Q: If f(x) is even, int[0,1]f(x)dx = 5, int[0,7]f(x)dx = 1, find...

[SharkeyC]

38. Suppose that f (x) is an even function, and let:

. . . . .\(\displaystyle \displaystyle \int_0^1\, \)\(\displaystyle f(x)\, dx\, =\, 5\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^7\, \)\(\displaystyle f(x)\, dx\, =\, 1\)

What is the value of the following?

. . . . .\(\displaystyle \displaystyle \int_{-7}^{-1}\, \)\(\displaystyle f(x)\, dx\)

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Dwindled it down to
-4
or
4
NEED HELP, this one problem is worth 25/50 points on my quiz.

 

[pka]

\(\displaystyle \large\displaystyle\int_{ - 7}^{ - 1} {f(x)dx} = \int_7^1 { - f( - x)dx} = \int_1^7 {f(x)dx} \)

Can YOU explain those steps?

 

[stapel]

38. Suppose that f (x) is an even function, and let:

. . . . .\(\displaystyle \displaystyle \int_0^1\, \)\(\displaystyle f(x)\, dx\, =\, 5\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^7\, \)\(\displaystyle f(x)\, dx\, =\, 1\)

What is the value of the following?

. . . . .\(\displaystyle \displaystyle \int_{-7}^{-1}\, \)\(\displaystyle f(x)\, dx\)

If a function is "even", what can you say about its y-values at equal distances from zero? In other words, what is the relationship between the values of y = f(-a) and y = f(+a)?

What are the rules for arithmetic on integrals? In particular, what are the rules for things like the following?

. . . . .\(\displaystyle \displaystyle \int_1^2\, \)\(\displaystyle g(x)\, dx\, +\, \)\(\displaystyle \displaystyle \int_2^3\, \)\(\displaystyle g(x)\, dx\)

. . . . .\(\displaystyle \displaystyle \int_1^3\, \)\(\displaystyle g(x)\, dx\, -\, \)\(\displaystyle \displaystyle \int_2^3\, \)\(\displaystyle g(x)\, dx\)

How can you relate this information to the posted exercise?