Question

Suppose that f(x) is an EVEN function and let the integral f(x) dx from 0 to 1=5 and the integral f(x) dx from 0 to 7 =1. What is the integral from f(x) dx from -7 to -1?

Answer 1

Since it's even, the integral from -7 to -1 is the same as the integral from 1 to 7.
think about
x^2 from 0..1,its 1/3
from 0..7 its 7^3/3
from -1..7 its 7^3/3+1/3

from -7..-1 of x^2 its -1/3+7^3/3 to get this without integrating we just do 7^3/3-1/3 which is the integral from 0..7 minus the integral from 0..1
So the integral from 1 to 0 f(x) dx = 5 changes to the integral from 0 to 1 f(x) dx = -5 and you add it to the integral from 0 to 7 which is 1 making -4

Answer 2

The integral of the even function f(x) from -7 to -1 is equal to -4.

How to get the integral?

Remember that an even function is a function such that:

f(-x) = f(x).

Also, remember that:

$\int\limits^a_b {g(x)} \, dx = -\int\limits^b_a {g(x)} \, dx$

Now we can rewrite:

$\int\limits^{-1}_{-7} {f(x)} \, dx = -\int\limits^{-7}_{-1} {f(x)dx}$

Because f(x) is even, we can change the negative signs:

$\int\limits^{7}_{1} {f(x)}dx$

This will be equal to:

$\int\limits^{7}_{1} {f(x)}dx = \int\limits^{7}_{0} {f(x)}dx - \int\limits^{1}_{0} {f(x)}dx$

And we know the two above ones, we will get:

$\int\limits^{7}_{1} {f(x)}dx = 1 - 5 = -4$

The integral is equal to -4.

If you want to learn more about integrals, you can read:

https://brainly.com/question/19053586

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