Blog Post #8

Identify Three Learning Targets

Learning Target Number 1: I can approximate the area under a curve using left and right endpoints.
     a.) You must first decide the width of the rectangles or the change in X of the function. This is a preference as to how accurate you wish to be when finding the area under the curve. The smaller the width or change in x, the more accurate the calculation will be as there are more rectangles.
     b.) The next step is to set up a summation to accurately represent the rectangles being used to calculate the area under the curve. The summation should include n= the number of rectangles, delta x= the width of the rectangles and which endpoint you will start with. (The summation with number of rectangles "n" above and i=1 below of delta x times the function of the lower limit plus i times delta x.) Or the summation of Delta X= (lower limit + I delta X)
     Ex. 4-2x^2 on [0,12] In this case, because I am not a fan of being accurate and I am lazy, I will have six triangles and my triangle widths will be two. Starting from the left endpoint, I will include zero and exclude twelve. Starting from the right endpoint, I will include twelve and exclude zero. From the left endpoint it will be 2 (f(0)+f(2)+f(4)+f(6)+f(8)+f(10))=-832
From the right endpoint it will be 2(f(2)+f(4)+f(6)+f(8)+f(10)+f(12))=-1408
What usually "trips me up the most" is knowing how accurate to be with my calculation. (Knowing how many rectangles to use for my area.)

Learning Target Number 2: I can find the area under the curve using the Riemman's Summation.
     a.) During Riemman's Summation, there is an infinite amount of rectangles. This is very similar to the previous learning target except that we use n as the number of rectangles, we take the problem as n approaches infinity, and delta x is the interval distance over n.
The limit as n approaches infinity of the summation of delta x times f(lower limit + i Delta x)
     b.) The rest of the process is exactly the same except there will be more reducing of summations. It is also essential to understand that the summation of i^2 is (2n^3 + 3n^2 + n)/6.
Ex. 4-2x^2 on [0,12]
1.) Take the limit as n approaches infinity of the summation of 12/n -2(0+12i/n)^2 + 4 2.) Distribute the square and then the negative two. 3.) Put the 12/n in front of the summation. 4.) Separate the i^2 from i^2/n^2 5.) Distribute the summation. 6.) Distribute the 12/n. 7.) Cancel the denominator of the summation of i^2. 8.) Disregard remaining terms containing "n" and simplify. 9.) The answer should be -1104.
The most difficult part for me when using Riemman's Summation is remembering the summations of i and i^2.

Learning Target Number 3: I can find the area under a curve using the fundamental theorem of calculus.
     a.) Find the anti derivative of the original function. Plug your upper value into the anti derivative and plug the lower value into the anti derivative. subtract the higher from the lower.
Ex. 4-2x^2 on [0,12]
4x-2/3x^3 + c is the anti derivative.
4(12)-2/3(12)^3 +c -  (4(0)-2/3(0)^3+c = -1104 -0 = -1104
The most difficult part when using the fundamental theorem of calculus part two is remembering which interval is first.