Inscribed Rectangles

Question

Which of the following values would be obtained using 10 inscribed rectangles of equal width (a lower sum) to estimate     ¹                                                                         ∫   ײdx                                                             °         
                                      

0.285

0.385

1.380

2.310

 

Answer 1

OK, so you are splitting f(x)=ײdx  into 10 rectangles of equal width over the range x=0 to x=1. The sum of the areas of those rectangles is an approximation of the integral of f(x). That's how the Fundamental Theorem of Calculus works, and how engineers use simulation programs like Matlab to do numerical integration. Figure that out now and you might have a nice career and make some money. LOL

But anyway. The ten rectangles of equal parts each have width of 0.1, since the range of integration is 1.

 

What you will need to do is find x² for each x in the range. Add those values up. They are the vertical sides of the ten rectangles. Then multiply by 0.1, and there is your approximation.

 

So I did this.

x = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1

x² = 0.01, 0.04, 0.09, 0.16, 0.25, 0.36, 0.49, 0.64, 0.81, 1

 

The sum of the vertical sides of the ten rectangles is 3.85.

 

And your approximation of the integral is equal to 0.385.

 

Sanity check: the definite integral is x³/3 as x goes from 0 to 1. So the definite integral is 1/3. That's pretty close, and good for an approximation.

 

But think about this: for each of the rectangles in the approximation, you used the right-hand vertical side to calculate with. What if you went with the left-hand side instead?

 

Your vertical sides would be

 

0, 0.01, 0.04, 0.09, 0.16, 0.25, 0.36, 0.49, 0.64, 0.81

 

Their sum would be 2.85 and your approximation 0.285. Dang. That's different from the other approximation you made. And that's one way you see that an approximation is not very precise. Someday you might learn better ones.

 

But for now, suppose you take the two approximations made here and got their average? That is 0.335, which is almost 1/3 exactly!

 

Cheers!