Finding Integrals

[burt]

>If \(f\) is a continuous, even function such that \(\displaystyle \int^3_0f(x)dx=-4\), then what is \(\displaystyle \int^3_{-3}5(f(x)+1)dx\)?

I was given this problem to solve, but I am not sure how to do it. How does the first integral help me find the second? What information does it give me?

 

[Deleted member 4993]

>If \(f\) is a continuous, even function such that \(\displaystyle \int^3_0f(x)dx=-4\), then what is \(\displaystyle \int^3_{-3}5(f(x)+1)dx\)?

I was given this problem to solve, but I am not sure how to do it. How does the first integral help me find the second? What information does it give me?

\(\displaystyle \int^3_{-3}5(f(x)+1)dx\)

=\(\displaystyle \int^3_{-3}5*f(x) dx \) + \(\displaystyle \int^3_{-3}5dx \)

continue......

 

[Dr.Peterson]

>If [MATH]f[/MATH] is a continuous, even function such that [MATH]\int^3_0f(x)dx=-4[/MATH], then what is [MATH]\int^3_{-3}5(f(x)+1)dx[/MATH]?

I was given this problem to solve, but I am not sure how to do it. How does the first integral help me find the second? What information does it give me?

I presume you can see that the two integrals are related.

The limits of integration have changed, in a way that relates to [MATH]f[/MATH] being an even function; and the integrand has been modified.

What properties of integrals do you know pertaining to adding or multiplying the integrand?

 

[pka]

>If \(f\) is a continuous, even function such that \(\displaystyle \int^3_0f(x)dx=-4\), then what is \(\displaystyle \int^3_{-3}5(f(x)+1)dx\)? I was given this problem to solve, but I am not sure how to do it. How does the first integral help me find the second? What information does it give me?

If \(\displaystyle g\) is even intergrable function then \(\displaystyle 2\int^3_0 g(x)dx=\int^3_{-3} g(x)dx\)

 

[burt]

If \(\displaystyle g\) is even intergrable function then \(\displaystyle \int^3_0 g(x)dx=2\int^3_{-3} g(x)dx\)

Shouldn't this be the opposite? The integral from 0 to 3 seems to be half of the integral from -3 to 3 not double?

 

[Deleted member 4993]

Shouldn't this be the opposite? The integral from 0 to 3 seems to be half of the integral from -3 to 3 not double?

You are correct.

 

[lookagain]

Shouldn't this be the opposite? The integral from 0 to 3 seems to be half of the integral from -3 to 3 not double?

Good! This is the type of correction that anyone should readily speak up and
offer as you did.

 

[pka]

>If \(f\) is a continuous, even function such that \(\displaystyle \int^3_0f(x)dx=-4\), then what is \(\displaystyle \int^3_{-3}5(f(x)+1)dx\)? I was given this problem to solve, but I am not sure how to do it. How does the first integral help me find the second? What information does it give me?

First I apologize for the typo that I have corrected. (Damn cut&past)
Here is an interesting question: If \(\displaystyle f\) is an even function then is \(\displaystyle g(x)=f(x)+1\) also an even function?
If yes then
\(\displaystyle \begin{align*}\int_{ - 3}^3 {5\left( {f(x) + 1} \right)dx} & = 2\int_0^3 {5\left( {f(x) + 1} \right)dx}\\& = 10\int_0^3 {\left( {f(x) + 1} \right)dx} \\& = 10\left[ {\int_0^3 {f(x)dx + \int_0^3 {1dx} } } \right]\\& = ?\end{align*}\)

 

[HallsofIvy]

I would presume that you know that \(\displaystyle \int_{-3}^3 5(f(x)+ 1)dx= 5\int_{-3}^3 f(x)dx+ 5\int_{-3}^3 dx= 5\int_{-3}^3 f(x)dx+ 5(6)= 5\int_{-3}^3 f(x)+ 30\)