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How do I find the exact area under the curve of the function $f(x)=4+3x-x^2$ on the interval $[-1,3]$ using integration. I'm super lost, and not quite sure how to start the problem.

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you just asked this question less than one hour ago...why again?! –  DonAntonio Jul 29 at 7:07
 
Do you know how to find the antiderivative (that is, indefinite integral) of a function? Or at the very least, do you know what those words mean? –  Omnomnomnom Jul 29 at 7:07
 
no this is a different question, i am only supposed to use limist and a riemann sum on the other one, and integration on this one. it is two different questions, not a duplicate. –  cschurman Jul 29 at 7:08
 
Do you know the fundamental theorem of calculus ? –  Tony Piccolo Jul 29 at 7:21
 
Do you know how to calculate $\int_{-1}^3 4 ~ dx$? How about $\int_{-1}^3 3x ~ dx$? –  Adriano Jul 29 at 7:26
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marked as duplicate by DonAntonio, iostream007, Omnomnomnom, Henry, Davide Giraudo Jul 29 at 8:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2 Answers

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Hints:

$$\forall\,1\ne n\in\Bbb N\;,\;\;\int x^ndx=\frac{x^{n+1}}{n+1}+C$$

$$\int\limits_a^bx^ndx=\left.\frac1{n+1}x^{n+1}\right|_a^b=\frac{b^{n+1}-a^{n+1}}{n+1}$$

$$\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$$

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okay, but i have no idea what numbers to plug in where, none of it makes any sense to me.... –  cschurman Jul 29 at 7:13
 
@cschurman, if that is true then you really need to approach your teacher and ask him directly, as otherwise what you're asking here is for someone else to do your work for you as you can't do any of it by yourself. This is not right and you won't learn that way. You need to go over the basic again and again until you grasp them. –  DonAntonio Jul 29 at 7:16
 
let me rephrase that, i understand exactly what you are saying, and i understand how that sounded... and its not what it sounded like! i get the concept of what i am needing to do, i just can't seem to figure out what exactly i need to do, i'm not asking for you to do my homework for me cause like you said i will not learn it that way, i'm just trying to get the help walking thru it so i do learn it. –  cschurman Jul 29 at 7:20
 
@cschurman, I've been a prof. mathematician the last 25 years +, and believe me: any more help on this question after your other question would already be the whole answer. Honest. –  DonAntonio Jul 29 at 7:23
 
oh okay. so as i run the limit on the previous question you helped me with, this one i need to find my upper and lower limits on the intergral, which the upper limit would be n which is the number of rectangles under the curve of the graph, and the lower limit would be i which is what you have marked as b and a respectively. am i correct on that? –  cschurman Jul 29 at 7:26
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$$\int_{-1}^{3}(4+3x+x^2)dx = \int_{-1}^{3}(4)dx + \int_{-1}^{3}(3x)dx+ \int_{-1}^{3}(x^2)dx$$ You should be able to solve those integrals rather easily.

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i should be able to but it will probably take me quite a bit of time to get them solved, as i have to look back thru my notes to figure which step is my next step. –  cschurman Jul 29 at 7:31
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