1. Applications Of Integrals

2. Rectangular Approximation n= number of rectangles F(x₁)= length of rectangle Δx= width of rectangle Note: Δx= (b-a) n Area = [f(x₁) + … + f(xn)] Δx Δx= difference in integral (b-a) # of squares

3. L-Ram (Over Approximation) Points are on the LEFT-handed side, on the curve Area = [f(x₁) + … + f(xn)] Δx

4. R-Ram (Under Approximation) Points are on the RIGHT-handed side, on the curve Area = [f(x₁) + … + f(xn)] Δx

5. M-Ram (In-Between Approximation) Points are in the MIDDLE of the curve, not on the curve Area = [f(x₁) + … + f(xn)] Δx

6. Calculator Shortcuts LRAM: sum(seq((f(x)Δx,x,a,b-Δx,Δx)) RRAM: sum(seq((f(x)Δx,x,a+Δx,b,Δx)) MRAM: sum(seq((f(x),Δx,x,a+Δx,b-Δx,Δx)) 2 2 equation width variable width beginning end

7. Which RAM Is Most Accurate? Out of LRAM, RRAM, and MRAM, MRAM is the most accurate It uses a combination of both LRAM and RRAM to get an in-between approximation instead of an over or under approximation If you have an area under the x-axis, compensate for the negative area by putting a – in front of the rectangular areas that fall below the x-axis

8. How Is Rectangular Approximation Linked To Area? Rectangular approximation is linked to area because it takes the area of a number of rectangles fitted under a curve to get an approximation to the real area. Using MRAM and a larger number of rectangles, the area you find will become closer to the actual area

9. Area Methods Area of a Triangle: AΔ= 1bh X-axis: A= Y-axis: A= Inverse Method: Switch x’s & y’s Solve for y Integral: Cone Method: A= 2

10. Volume Methods Disk Method: X-Axis: Y-Axis: same as the X-Axis, only change the boundaries and equation to terms of y Washer Method: Shell: Useful for y-axis rotation Same thing as: dx/dy