Evaluate the definite integral of
(4x^3-x^2+2) from 5 to 2.
Can someone please explain what I'm being asked to do? My book is not clear on this concept.
First of all, the integral and the derivative are inverse operations
e.g.
if y = 4x^3
then dy/dx = 12x^2
then ∫ 12x^2 dx = 4x^3
notice the notation ,
read it as "the integral of 12x^2 by dx "
if we want the definite integral then usually you will find two numbers with the ∫ integral sign, the smaller number below it, and a larger number above it. (I can't type it here)
You would then substitute, and get
(value of the integral using the upper value) - (value of the integral using the lower value)
so for your question.....
∫ ( 4x^3 - x^2 + 2) dx from x = 2 to 5
= [ x^4 - (1/3)x^3 + 2x] from 2 to 5
= (5^4 - (1/3)5^3 + 2(5) ) - (2^4 - (1/3)(2^3) + 2(2) )
= 625 - 125/3 + 10 - 16 + 8/3 - 4
= 615 - 117/3
= 615- 39
= 576
(check my arithmetic, I am prone to errors today)
To see a geometrical interpretation of what we did, do the following
go to
http://rechneronline.de/function-graphs/
in the "first graph" window enter:
4x^3 - x^2 + 2 , (type it exactly that way)
in 'Range x-axis from' enter 2 and 5
in 'Range y-axis from' enter 0 and 500
click on "Draw"
What our answer of 576 represents is the area between the curve and the x-axis from x = 2 to x = 5
(our answer of 576 is reasonable if we consider the
average height of the "triangle" to be (477+30)/2= appr 254 and our base is 3, from 2 to 5
254x3 = 762
of course this answer is too large, since we joined the endpoints with a straight line)
Oh wow. Thank you so much. That was much more help than the book gave me! Thanks and God bless, Reiny!
You are welcome
Certainly! In this problem, you are being asked to evaluate the definite integral of a given function over a certain interval. Let's break it down step by step:
1. Integration: Integration is a mathematical operation that is the reverse of differentiation. It allows you to find the antiderivative of a function. In this problem, the function you need to integrate is (4x^3 - x^2 + 2).
2. Definite integral: A definite integral is a type of integral where you find the area under the curve of a function over a specific interval. In this problem, you need to evaluate the definite integral of the given function from 5 to 2.
3. Given interval: The interval from 5 to 2 represents the range of x values over which you will find the area under the curve.
To evaluate the definite integral, you can follow these steps:
Step 1: Find the antiderivative of the function. To do this, integrate each term individually. The antiderivative of 4x^3 is (4/4)x^4 = x^4, the antiderivative of -x^2 is (-1/3)x^3, and the antiderivative of 2 is 2x. So, the antiderivative of the given function is x^4 - (1/3)x^3 + 2x.
Step 2: Evaluate the antiderivative at the upper limit of the interval (in this case, at x = 2). Substitute x = 2 into the antiderivative expression, so you have (2^4 - (1/3)(2^3) + 2(2)) = 16 - (8/3) + 4.
Step 3: Evaluate the antiderivative at the lower limit of the interval (in this case, at x = 5). Substitute x = 5 into the antiderivative expression, so you have (5^4 - (1/3)(5^3) + 2(5)) = 625 - (125/3) + 10.
Step 4: Calculate the difference between the two values obtained in Step 2 and Step 3. (16 - (8/3) + 4) - (625 - (125/3) + 10) = 20 - (125/3) = 375/3 - 125/3 = 250/3.
Therefore, the definite integral of (4x^3 - x^2 + 2) from 5 to 2 is equal to 250/3.