# Help W/ Definite Integrals

posted by .

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.

• Help W/ Definite Integrals -

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)

∫ ( 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)

• Help W/ Definite Integrals -

Oh wow. Thank you so much. That was much more help than the book gave me! Thanks and God bless, Reiny!

• Help W/ Definite Integrals -

You are welcome

## Similar Questions

1. ### Calculus

Solve these indefinite and definite integrals. [integration sign] 4j-5j^3 dj I got 2j^2 - 5/4j^4... is this my final answer?
2. ### calculus

There are four integrals: 1) definite integral x/(1+x^4)dx b/w 0_infinity 2) definite integral (x^2)/(1+x^4)dx b/w 0_infinity 3) definite integral (x^3)/(1+x^4)dx b/w 0_infinity 4) definite integral (x^4)/(1+x^4)dx b/w 0_infinity Which …
3. ### calculus

There are four integrals: 1) definite integral x/(1+x^4)dx b/w 0_infinity 2) definite integral (x^2)/(1+x^4)dx b/w 0_infinity 3) definite integral (x^3)/(1+x^4)dx b/w 0_infinity 4) definite integral (x^4)/(1+x^4)dx b/w 0_infinity Which …
4. ### Calculus

Can someone explain to me how to do these?
5. ### Calc

If we know that the definite integral from -6 to -3 of f(x) equals 6, the definite integral from -6 to -5 equals 2 and the definite integral from -4 to -3 equals 4 then: What is the definite integral from -5 to -4?
6. ### Calculus (urgent help)

consider the function f that is continuous on the interval [-5,5] and for which the definite integral 0(bottom of integral sign) to 5(top of integral sign) of f(x)dx=4. Use the properties of the definite integral to evaluate each integral: …
7. ### calculus

consider the function f that is continuous on the interval [-5,5] and for which the definite integral 0(bottom of integral sign) to 5(top of integral sign) of f(x)dx=4. Use the properties of the definite integral to evaluate each integral: …
8. ### calculus (please with steps and explanations)

consider the function f that is continuous on the interval [-5,5] and for which the definite integral 0(bottom of integral sign) to 5(top of integral sign) of f(x)dx=4. Use the properties of the definite integral to evaluate each integral: …
9. ### calculus

Evaluate lim (1³ +2³ +3³ +…+ n3)/n^4 n →∞ by showing that the limit is a particular definite integral and evaluating that definite integral.
10. ### Math (Definite Integrals)

Sketch the region given by the definite integral. Use geometric shapes and formulas to evaluate the integral (a > 0, r > 0). r ∫ sqrt(r^2 - x^2) dx -r While I recognize that this looks similar to a circle function, I'm not …

More Similar Questions