Area approximation and (riemann?) sums

In summary, the student is a first-year physics student learning calculus and has a question about approximating the area of a region bounded by y = 0. They have already solved the first part of the problem, finding the area bounded by y = 5/x, and are now wondering about the area bounded by y = 0. The answer to this is zero, and the student has mistakenly thought it was a separate problem. The person responding mentions that the area bounded by y = 5/x and y = 0 is actually infinite, not 10.4 as the student estimated.
  • #1
science_rules
158
2

Homework Statement


I am a first-year physics student learning calculus. my question is about the approximation of the area of a region bounded by y = 0.

Homework Equations


Use rectangles (four of them) to approximate the area of the region bounded by y = 5/x (already did this one), and y = 0. The height of the tallest rectangle is 5 units, on the y-axis. The numbers(width of the rectangles) on the x-axis are respectively: 1, 2, 3, 4 units.


The Attempt at a Solution


The first question was to find the area bounded by y = 5/x which i did: 5/1 + 5/2 +5/3 + 5/4 = approximately 10.4. I know this is correct.
Then the area bounded by y = 0 would just be zero i assume? is there a certain way to write this one or is it just zero? i cannot see what else it could be.
 
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  • #2
I think they just mean to say find the area bounded BETWEEN the two curves y=5/x and y=0. It's just a single problem, which you already did. Not two separate problems.
 
  • #3
ohhh. good to know. silly me. thankyou
 
  • #4
science rules,
You might find it interesting that your estimate of 10.4 for the area bounded by the graph of y = 5/x and the x-axis (the line y = 0) is a very low estimate. In fact, the area of this region is infinite.
 

Related to Area approximation and (riemann?) sums

1. What is area approximation?

Area approximation is a method used in mathematics to estimate the area of a shape or region. It involves breaking down the shape into smaller, simpler shapes and calculating their areas, then adding them together to get an approximate total area.

2. How is area approximation useful?

Area approximation is useful because it allows us to find the area of irregular or complex shapes that cannot be easily measured or calculated using traditional methods. It also provides a way to estimate the area of a shape without needing precise measurements.

3. What is a Riemann sum?

A Riemann sum is a type of area approximation technique that uses rectangles to estimate the area under a curve on a graph. The width of the rectangles is determined by the interval size and the height is determined by the function at a specific point within the interval.

4. How is a Riemann sum calculated?

To calculate a Riemann sum, you need to divide the interval of the function into smaller subintervals and determine the width of each subinterval. Then, you find the height of the rectangle for each subinterval by plugging in a value from that subinterval into the function. Finally, you add all the areas of the rectangles together to get an approximation of the total area under the curve.

5. What is the difference between a left, right, and midpoint Riemann sum?

A left Riemann sum uses the left endpoint of each subinterval to determine the height of the rectangle, a right Riemann sum uses the right endpoint, and a midpoint Riemann sum uses the midpoint of each subinterval. The choice of endpoint or midpoint can affect the accuracy of the approximation, with a midpoint Riemann sum typically being the most accurate.

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