Evaluating Integrals: ∫_(-4)^1[f(x)]dx = (-125)/6

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In summary, the notation ∫_(-4)^1[f(x)]dx represents the definite integral of the function f(x) from the lower limit of -4 to the upper limit of 1, which is the area under the curve of the function between these limits. To evaluate this integral, you would find the anti-derivative of the function and substitute the limits into it. The value (-125)/6 represents the numerical value of the definite integral, which can be evaluated using a calculator but it is important to understand the steps involved. Evaluating integrals has many real-world applications, such as calculating area, distance, volume, and probabilities.
  • #1
Noah1
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Can anyone tell me if this is correct?
\int
∫1 at top and -4 on bottom of symbol [1^3/3+3 1^2/2-(4 x 1)+C] - [〖-4〗^3/3+〖3x(-4)〗^2/2-(4 x-4)+C]
If f(x) = x^2+3x-4, then F(x) = x^3/3+3 x^2/2-4x+C
∫_(-4)^1[1^3/3+3 1^2/2-(4 x 1)+C] - [〖-4〗^3/3+〖3x(-4)〗^2/2-(4 x-4)+C]
[1/3+3/2-4+C] - [-64/3+24+16+C]
[11/6-4+C] - [64/3-40+C]
11/6-4+C +64/3-40-C
= 139/6-44
= 139/6-(-264)/6
= (-125)/6
 
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  • #2
Hi Noah. :D

It's correct. As with any definite integral, you may leave out the constant of integration (C).
 
  • #3
thank you
 

FAQ: Evaluating Integrals: ∫_(-4)^1[f(x)]dx = (-125)/6

1. What does the notation ∫_(-4)^1[f(x)]dx mean?

The notation ∫_(-4)^1[f(x)]dx represents the definite integral of the function f(x) from the lower limit of -4 to the upper limit of 1. In other words, it represents the area under the curve of the function f(x) between the x-values of -4 and 1.

2. How do you evaluate this integral?

To evaluate this integral, we would first need to find the anti-derivative of the function f(x). Once we have the anti-derivative, we can substitute the upper and lower limits into the anti-derivative and take the difference to find the value of the definite integral.

3. What does the value (-125)/6 represent in this integral?

The value (-125)/6 represents the numerical value of the definite integral evaluated between the limits of -4 and 1. In other words, it is the area under the curve of the function f(x) between the x-values of -4 and 1.

4. Can you use a calculator to evaluate this integral?

Yes, you can use a calculator to evaluate this integral. Many calculators have a built-in integral function that can be used to evaluate definite integrals like this one. However, it is important to understand the steps involved in evaluating an integral by hand in order to fully understand the concept.

5. What are some real-world applications of evaluating integrals?

Evaluating integrals has many real-world applications, including calculating the area under a curve in physics or engineering problems, finding the total distance traveled by an object, and determining the volume of an irregularly shaped object in mathematics and engineering. It is also commonly used in economics and statistics to calculate probabilities and expected values.

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