Equating two integrals with a constant involved

[Arnoldjavs3]

Homework Statement

$\int_{-1}^{3} f(x) dx = -4 = \int_{-1}^{3} 2g(x)dx$

Now find a value(constant a) that makes the following true:

$\int_{-1}^{3} [3f(x) - ag(x) +a] dx = \int_{-1}^{3}(1-ax)dx$

Homework Equations

The Attempt at a Solution

I'm unsure if my approach here is correct but I think that I need to utilize the fact that $\int_{-1}^{3} f(x) dx$ = -4 and substitute this into the left side of equation? So I would get something like this after evaluating:

$\int_{-1}^{3} [-12+2a+a] dx$ and then once I've done this, i can solve the integral. I'll have to solve the integral on the right side as well and once I've finished that, I find a value for a? I got 52/16 but I don't have an answer to check to.

 

[Mark44]

Homework Statement

$\int_{-1}^{3} f(x) dx = -4 = \int_{-1}^{3} 2g(x)dx$

Now find a value(constant a) that makes the following true:

$\int_{-1}^{3} [3f(x) - ag(x) +a] dx = \int_{-1}^{3}(1-ax)dx$

Homework Equations

The Attempt at a Solution

I'm unsure if my approach here is correct but I think that I need to utilize the fact that $\int_{-1}^{3} f(x) dx$ = -4 and substitute this into the left side of equation?

You can also use the fact that $\int_{-1}^{3} 2g(x)dx = -4$ to get an expression for $\int_{-1}^{3} g(x)dx$.

So I would get something like this after evaluating:

$\int_{-1}^{3} [-12+2a+a] dx$ and then once I've done this, i can solve the integral. I'll have to solve the integral on the right side as well and once I've finished that, I find a value for a? I got 52/16 but I don't have an answer to check to.

The two properties of definite integrals that come into play here are:
$\int_a^b k \cdot f(x) dx = k\int_a^b f(x) dx$ (k a constant)
and $\int_a^b f(x) + g(x) dx = \int_a^b f(x) dx + \int_a^b g(x) dx$

 

[Arnoldjavs3]

You can also use the fact that $\int_{-1}^{3} 2g(x)dx = -4$ to get an expression for $\int_{-1}^{3} g(x)dx$.The two properties of definite integrals that come into play here are:
$\int_a^b k \cdot f(x) dx = k\int_a^b f(x) dx$ (k a constant)
and $\int_a^b f(x) + g(x) dx = \int_a^b f(x) dx + \int_a^b g(x) dx$

using these properties would I get something like this?(after splitting the left side):
$3\int_{-1}^{3} f(x)dx - a\int_{-1}^{3} g(x)dx + \int_{-1}^{3} adx$ ?

 

[Mark44]

using these properties would I get something like this?(after splitting the left side):
$3\int_{-1}^{3} f(x)dx - a\int_{-1}^{3} g(x)dx + \int_{-1}^{3} adx$ ?

Yes, and you should be able to evaluate all three of these integrals, based on the given information.