Evaluate S8.4.3.84 Integral from 1 to 3

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In summary, the given integral can be evaluated by breaking it down into three separate integrals and then combining them into one using the properties of integration. The interval for the integral is $y \in [1,3]$ and there is no division by zero. The final result for the integral is $\ln(3)$.
  • #1
karush
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$\tiny{s8.4.3.84}$
Evaluate $I=\displaystyle\int_1^3 \dfrac{y^3-2y^2-y}{y^2}\ dy$
\begin{array}{lll}\displaystyle
\textit{expand}
&I=\displaystyle\int_1^3 y \ dy
-\displaystyle\int_1^3 2 \ dy
-\displaystyle\int_1^3 \dfrac{1}{y} \ dy
\end{array}

just want to see if I am on the right horse before I cross the stream:cool:
 
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  • #2
interval is $y \in [1,3]$

no division by zero, so it’s all good ... but why not just one definite integral to evaluate

$\displaystyle \int_1^3 y - 2 - \dfrac{1}{y} \, dy$

instead of three?
 
  • #3
skeeter said:
interval is $y \in [1,3]$

no division by zero, so it’s all good ... but why not just one definite integral to evaluate

$\displaystyle \int_1^3 y - 2 - \dfrac{1}{y} \, dy$

instead of three?
well don't we have to integrate them individually anyway

integrate
$I=\biggr|\dfrac{y^2}{2}-2y-\ln{y}\biggr|_1^3$
calculate
$I=4-4-\ln (3)=\ln(3)$

this was an even problem number so no book answer
 
  • #4
you really need to pay attention to your signs ...

$-\ln(3) = \ln\left(\dfrac{1}{3}\right)$
 

FAQ: Evaluate S8.4.3.84 Integral from 1 to 3

1. What is the purpose of evaluating S8.4.3.84 Integral from 1 to 3?

The purpose of evaluating this integral is to find the exact numerical value of the area under the curve of the function S8.4.3.84 between the limits of 1 and 3. This can provide valuable information about the behavior of the function and its relationship to other functions.

2. How is the integral from 1 to 3 calculated?

The integral from 1 to 3 is calculated using a mathematical process called integration, which involves breaking down the function into smaller parts and finding the sum of these parts. This can be done using various techniques such as the Riemann sum or the fundamental theorem of calculus.

3. What factors can affect the value of the integral from 1 to 3?

The value of the integral can be affected by several factors, such as the shape of the function, the limits of integration, and any discontinuities or singularities in the function. It can also be affected by the method used to evaluate the integral and the precision of the calculations.

4. How can the accuracy of the integral from 1 to 3 be determined?

The accuracy of the integral can be determined by comparing the calculated value to the exact value, if known. This can be done using techniques such as error analysis or by using more precise methods of integration. Additionally, the integral can be approximated using numerical methods to determine its accuracy.

5. What is the significance of the integral from 1 to 3 in scientific research?

The integral from 1 to 3 can be used to solve a variety of real-world problems in fields such as physics, engineering, and economics. It can also provide insights into the behavior and relationships of mathematical functions, which can be applied to further research and developments in various scientific fields.

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