Integrals

Defining an Integral:
means the area between f(x) and the x-axis between where x=a and x=b. Integrals represent the overall total or accumulation of distance between two points in a function. 'a' represents the left limit, 'b' represents the right limit, f(x) is the function and dx is the width. (Before learning integrals, we were introduced to Riemann Sums which acted as small rectangles of an infinite width from a function to the x-axis and adding up the areas. Integrals are exact while Riemann Sums are more general depending on how small or large the rectangle widths are.)

General Formula for an Integral:
To calculate this area you must do the anti-derivative of the function expressed as F and plug in your upper limit for F(x) then subtract F(x) with your lower limit plugged in for x as represented here.

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Example: By using rule #1 we can anti-derive this . f(x)= x
If you are looking to find the average value of a function you use the anti-derivative to come up with this value. You can double check by making sure when you multiply your exponent with the coefficient, you have the same value as the coefficient in your original equation. Two (the exponent), multiplied by 1/2 (the coefficient) gives you one, which was the original exponent in the equation f(x) = x.

Graphing of integrals:
The interval that would represent this graph would be. When finding the area or the integral of f(x) graphically, you would find the area of the function between the graph and the x-axis. If the area is above the x-axis, the area is positive and if the area is below the x-axis, the area is negative. You would add up all of the separate areas you calculated and your answer would be equal to the anti-derivative of that function between two x-values.

Practice Problems:
First, you would use your rule of and you get from x=1 to x=2. Then you plug in your x-values, first x=2 then subtract the function with the x value of 1 plugged in.
 * 1) 1. No calculator section. Evaluate the following integral:

Answer: 6

 * 1) 2. Calculator Available:

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Anti-derivative is from x = 2 to x=? is equal to 10. (1/3 (x^3) - 1.5(x^2)) - (1/3 (2^3) - 1.5 (2^2)) = 10 1/3(x^3) - 1.5(x^2) + 3.333 = 10 - 3.333 -3.333 1/3(x^3) - 1.5(x^2) = 20/3 y1 = 20/3 y2 = 1/3(x^3) - 1.5(x^2) Trace and see where the two graphs intersect: 5.2309253 Answer: x = 5.231

Multiple Choice Question:
If

e. 7b - 6a
The Answer is choice C. because the integral of f(x) + 5 is the function a + 2b plus the integral from a to b of 5x. 5(b) - 5(a) is added together with the integral of f(x). a + 2b + 5b - 5a = 7b - 4a.