Definite Integral: Evaluate ∫1^8 (3 - y^(1/3) / y^(2/3)) dy | Problem #24"

In summary, a definite integral is a mathematical concept used to measure the area under a curve between two points on a graph. It is represented by the symbol ∫ and is used to find the exact value of the area between the curve and the x-axis. The notation ∫1^8 (3 - y^(1/3) / y^(2/3)) dy represents the definite integral of the function (3 - y^(1/3) / y^(2/3)) with respect to the variable y, evaluated from 1 to 8. To evaluate a definite integral, you must first find the indefinite integral of the given function and then substitute the upper and lower limits into it. The steps for evaluating the definite integral
  • #1
rowdy3
33
0
Evaluate each Definite integral.
∫ lower limit 1, upper limit 8. 3 - y^(1/3) / y^(2/3) ; dy
Here's a link if you have trouble understanding my problem. It's #24.
http://pic20.picturetrail.com/VOL1370/5671323/23643016/396407299.jpg
I did
∫(3*y^(-2/3) - y^(-1/3)) dy from 1 to 8

9*y^(1/3) - 3/2*y^(2/3) eval. from 1 to 8

= 9(2) - 6 - 9 + 3/2

= 3 + 3/2 = 9/2
 
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  • #2
Greetings! Good work, your answer is correct.
 

Related to Definite Integral: Evaluate ∫1^8 (3 - y^(1/3) / y^(2/3)) dy | Problem #24"

What is a definite integral?

A definite integral is a mathematical concept used to measure the area under a curve between two points on a graph. It is represented by the symbol ∫ and is used to find the exact value of the area between the curve and the x-axis.

What does the notation ∫1^8 (3 - y^(1/3) / y^(2/3)) dy represent?

This notation represents the definite integral of the function (3 - y^(1/3) / y^(2/3)) with respect to the variable y, evaluated from 1 to 8. In other words, it is finding the area under the curve of the given function between the points 1 and 8 on the y-axis.

How do you evaluate a definite integral?

To evaluate a definite integral, you must first find the indefinite integral of the given function. Then, substitute the upper and lower limits into the indefinite integral and subtract the result of the lower limit from the result of the upper limit. This will give you the numerical value of the definite integral.

What are the steps for evaluating the definite integral ∫1^8 (3 - y^(1/3) / y^(2/3)) dy?

The steps for evaluating this definite integral are as follows:

  1. Find the indefinite integral of the given function: ∫(3 - y^(1/3) / y^(2/3)) dy = 3y - 3(y^(1/3))/2 + C
  2. Substitute the upper and lower limits into the indefinite integral: 3(8) - 3(8^(1/3))/2 - (3(1) - 3(1^(1/3))/2)
  3. Simplify the expression: 24 - 3(2)/2 - (3 - 3(1)/2)
  4. Compute the final result: 24 - 3 - 3 + 3/2 = 18.5

What is the significance of evaluating a definite integral?

Evaluating a definite integral allows us to find the exact numerical value of the area between a curve and the x-axis. This can be useful in a variety of applications, such as calculating displacement, velocity, and acceleration in physics, or finding the total cost or profit in economics.

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