Estimate the area under the graph of f(x) =x^2 + 4x from x=1 to x=9

In summary, estimating the area under the graph of a function involves finding an approximation of the total area within a given range. This can be done using Riemann Sum, where the area is divided into smaller rectangles and their individual areas are added together. In this problem, the limits of integration are x=1 and x=9, meaning the area between those x-values is being calculated. The number of rectangles used in Riemann Sum can vary, with a higher number leading to a more accurate estimate. Other methods, such as the Trapezoidal Rule or Simpson's Rule, can also be used to solve this problem, but Riemann Sum is often the first method taught in introductory calculus courses.
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Estimate the area under the graph of f(x) =x^2 + 4x from x=1 to x=9 using 4 approximating rectangles and left endpoints.

first i had to find delta x, so i did 9-1/4 = 2
which means x0 = 0, x1 = 2, x2= 4, x3=6 (since I'm using left endpoints, i include x0)

after that, i just plug it in the left end point formula:
L4 = 2*0 + 2*12 + 2*32 + 2*60 = 208

i done the calculations many time and get the same answer, am i missing something?
 
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  • #2
the x's should be 1,3,5,7... you started at 1
 
  • #3
ah, i see. thanks for the help.
 

1. What does it mean to "estimate the area under the graph"?

Estimating the area under the graph of a function means finding an approximation of the total area between the graph and the x-axis within a given range.

2. How do you find the area under a curve using the given function?

To find the area under the graph of a function, you can use a method called Riemann Sum. This involves dividing the area into smaller rectangles and finding their individual areas, then adding them together to get an estimate of the total area.

3. What are the limits of integration in this problem?

The limits of integration in this problem are x=1 and x=9. This means we are finding the area under the graph between the x-values of 1 and 9.

4. How do you choose the number of rectangles for Riemann Sum?

The number of rectangles used for Riemann Sum can vary depending on the desired level of accuracy. Generally, the more rectangles used, the more accurate the estimate will be. A common approach is to start with a small number of rectangles and gradually increase it until the desired level of accuracy is achieved.

5. Can this problem be solved using other methods besides Riemann Sum?

Yes, there are other methods that can be used to estimate the area under a curve, such as the Trapezoidal Rule or Simpson's Rule. However, Riemann Sum is commonly used and is often the first method taught in introductory calculus courses.

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