Equating two integrals with a constant involved

• Arnoldjavs3
Then you can solve for a by setting the resulting expression equal to the integral on the right side and solving for a.
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##

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.

Arnoldjavs3 said:

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##

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##.
Arnoldjavs3 said:
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##

Mark44 said:
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## ?

Arnoldjavs3 said:
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.

Arnoldjavs3

1. Can you explain the process of equating two integrals with a constant involved?

The process of equating two integrals with a constant involved involves setting the two integrals equal to each other and solving for the unknown constant. This can be done by using algebraic manipulation and techniques such as substitution or integration by parts.

2. Why is it important to equate two integrals with a constant involved?

Equating two integrals with a constant involved allows us to find a relationship between two different functions and determine the value of the unknown constant. This can be useful in solving problems involving multiple variables or in finding the area under a curve.

3. What is the difference between equating two integrals with a constant involved and setting two functions equal to each other?

Equating two integrals with a constant involved involves setting the entire integral expression equal to another integral expression, including the constant. Setting two functions equal to each other involves equating only the functions themselves, without any integration involved.

4. Are there any specific rules or guidelines to follow when equating two integrals with a constant involved?

There are no specific rules or guidelines, but it is important to ensure that the integrals are set up correctly and that any constants are properly accounted for. It is also helpful to have a good understanding of algebraic manipulation and integration techniques.

5. Can you provide an example of equating two integrals with a constant involved?

Sure, for example, we can equate the integrals of sin(x) and 2cos(x) by setting ∫sin(x)dx = ∫2cos(x)dx. By using the double-angle identity, we can then solve for the constant and determine that ∫sin(x)dx = -2sin(x) + C.

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