# Please check if i am right - definite inegral

• DemiMike
In summary, the question asks for the use of the definition of definite integral with the right hand rule to evaluate a given integral. The work involves taking the limit as n approaches infinity and using summation formulas to simplify the expression. The result is found to be 72.5, but the use of a shortcut method is not allowed.
DemiMike
The question:
Use the definiton of the definite inegral (with right hand rule) to evaluate the following integral. Show work please
Can NOT use shortcut method.. must be the long process

Function:
1
S (3x^2 - 5x - 6) dx
-4

Work:
∫[-4,1] (3x^2 - 5x - 6) dx =
lim[n-->∞] 5/n ∑[i=1 to n] {3(-4 + 5/n)² - 5(-4 + 5/n) - 6} =
lim[n-->∞] 5*∑[i=1 to n] (48/n - 120i/n² + 75i²/n³ + 20/n -25i/n² - 6/n) =
lim[n-->∞] 5*∑[i=1 to n] (62/n - 145i/n² + 75i²/n³) =
lim[n-->∞] 5[62n/n - 145n(n+1)/(2n²) + 75n(n+1)(2n+1)/(6n³)] =
5(62 - 145/2 + 25) = 72.5

∑[i=1 to n] 1 = n
∑[i=1 to n] i = n(n+1)/2
∑[i=1 to n] i² = n(n+1)(2n+1)/6

Please check if this is correct and let me know

That looks ok to me. Even if you can't use the 'shortcut method' to solve the problem, you can always use the shortcut method to check your work.

it looks ok to me to =P, but i wanted to make sure if it's completely right

-thanks =)
which shortcut formula would i have to use

DemiMike said:
it looks ok to me to =P, but i wanted to make sure if it's completely right

-thanks =)
which shortcut formula would i have to use

I was assuming the 'shortcut' method was finding the antiderivative and evaluating it between the two limits. If you haven't learned that yet, then never mind.

## What is a definite integral?

A definite integral is a mathematical concept used to find the area under a curve or the net value of a function over a specific interval. It is represented by the symbol ∫ and has a lower and upper limit.

## How is a definite integral different from an indefinite integral?

An indefinite integral is a mathematical concept that represents the set of all possible antiderivatives of a function. It does not have a specific interval, unlike a definite integral, which has a defined upper and lower limit.

## What is the process for solving a definite integral?

To solve a definite integral, you need to determine the limits of integration, find the antiderivative of the function, and then substitute the upper and lower limits into the antiderivative. The resulting value is the definite integral.

## How is a definite integral used in real life?

Definite integrals are used in various fields of science and engineering, such as physics, economics, and statistics, to model and solve real-world problems. For example, in physics, definite integrals are used to calculate the work done by a force or the distance traveled by an object.

## What are some common applications of definite integrals?

Some common applications of definite integrals include finding the area or volume of irregular shapes, calculating displacement and velocity in kinematics, and determining the average value of a function over a specific interval.

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