Solve Integrals: Getting the Approximated Area

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Studying integrals...

I am studying for my maths B exam that is coming up and may need some help in understanding some concepts.

Here is one question that i was getting some practise on but i got the wrong answer !

Q: The area under the curve x²+ 1 between 1 and 4 is approximated by left rectangles of width 1 unit. What is the total area of the aproximation?

A: Well I calculated the area using the fundamental theorem of calculus and I ended up with 24...but the answer was wrong it turned out to be 17 according to BOB. I then remember that the fundamental theorem of calculus give you the exact area not the aproximated area. I am not so sure about how to get the approximated area though. Is there a formula or something I can use?

I know that the x and y values I am interested in are as follows (0, 1), (1, 2), (2, 3) and (3, 4). I am trying to find the area under the curve at these points. I drew in the rectangles under the curve and found the area but I got a value of 10 . Nothing is working for me at the moment so I need help...I really did think that that would work but of course 1 + 2 + 3 + 4 = 10 . But i haven't tried making the rectangles in smaller subdivisions yet so maybe that's what I have to do...and if anyone knows a formula or an easier way please tell me!

Well I'll try the smaller subdivision thing now and see if I get 17...and if I don't i'll be a little , but if someone has an easier way or some kind of formula i'll be

 

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I think I'm on the wrong track some how

 

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Don't worry I figured it out...I was interpereting it wrongly, well actually the method I was using was wrong . But its all good now . I worked it out my self yay! Now how did I get so confused on one of the easier questions...this always happens to me all the more difficult things i get but the easy things i don't, I am so weird sometimes.

 

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How did I do it?

For those of you that are interested this is what i did:

Well I was thinking about the properties of a quadratic equation and then it came to me...I have to square the x values to get a y value (I don't know how I got this wrong maybe I was a little overwhelmed by the scary word integral ) Any way I decided to draw the graph accurately and then draw rectangles under the curve, I had to work out the area of these rectangles to get the estimated area under the curve. Any way back to the equation, the equation was x² + 1, I knew that the y values are equal to x² + 1 (obviously ). The area under the curve I was interested in was below these points (1,3), (2, 5) and (3,4).

Here is an example of what I mean by rectangles under the curve: http://i122.photobucket.com/albums/o272/science_f/000000000000000000rectanglesundercu.jpg Sorry about how crappy the drawing looks but you should get what I mean.

I calculated the area of the rectangles using L x W = A after I had worked out what all of the areas were I added them all to gether to get the estimated area. I ended up with 2 + 5 + 10 = 17. And there you have it the correct answer 17 .