Solve Integrals: Getting the Approximated Area

In summary: So if you are having trouble understanding integrals or just need some help getting started I suggest you look into studying integrals using this method. In summary, Studying integrals can be difficult, but this method can help you understand the concepts better.
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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 :eek:!

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 :grumpy:. 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 :redface:. 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 :mad:, but if someone has an easier way or some kind of formula i'll be :biggrin:
 
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I think I'm on the wrong track some how :confused:
 
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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 :blushing:. But its all good now :cool:. I worked it out my self :approve: yay! :rofl: 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. :biggrin:
 
  • #4
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 :blushing: 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 :rolleyes:). 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 :redface: 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 :biggrin:.
 
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1. What is an integral?

An integral is a mathematical concept used to calculate the area under a curve. It is represented by the symbol ∫ and is used to find the area between a function and the x-axis.

2. How do you solve integrals?

To solve an integral, you must first determine the limits of integration, which are the upper and lower bounds of the area you want to find. Then, you can use a variety of integration techniques such as substitution, integration by parts, or trigonometric substitution to find the antiderivative of the function. Finally, you can evaluate the antiderivative at the limits of integration to find the approximated area.

3. What is the purpose of approximating the area under a curve?

The purpose of approximating the area under a curve is to find an estimate of the actual area, particularly when the function is complex and cannot be easily integrated. This can be useful in various fields such as physics, engineering, and economics where finding precise areas can help in making predictions and solving problems.

4. What are some common methods for approximating integrals?

Some common methods for approximating integrals include the midpoint rule, trapezoidal rule, and Simpson's rule. These methods use different formulas and techniques to divide the area under the curve into smaller, simpler shapes and then calculate the sum of their areas to find an approximation for the integral.

5. How accurate are approximated integrals?

The accuracy of an approximated integral depends on the method used and the number of subdivisions used to calculate the sum of the smaller areas. Generally, the more subdivisions used, the more accurate the approximation will be. However, it is important to note that an approximated integral will never be as accurate as the actual value, but it can provide a close estimate for practical purposes.

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