The document discusses various methods for approximating the area under a curve including: - Left, right, and midpoint rectangular approximation methods which divide the area into rectangles to estimate it - Area methods involving integrating curves bounded by the x-axis, y-axis, or between two curves - Volume methods like the disk, washer, and shell methods that revolve curves around axes to find volumes and relate to areas It concludes that the rectangular approximation methods are used to estimate areas by dividing them into rectangles, and that the midpoint method provides the most accurate approximation.

1. Group 4 (Alex Gagliardi) Applications of Integration

2. Area Under a Curve: Rectangular Approximation

3. Left Rectangular Approx. Method (LRAM) Right Rectangular Approx. Method (RRAM) Midpoint Rectangular Approx. Method (MRAM n= number of rectangles Delta x= width of rectangle in the interval [a,b] MRAM is the most accurate because the area that you are finding is more accurate compared to RRAM and LRAM.

4. Area Methods

5. X-axis Original equations: y= Left Right In terms of … Two curves: top bottom One curve below X-axis: -

6. Y-axis Original Equations Y- axis Bottom Top In terms of y: … Outer – inner /or right – left One curve on left y-axis:

7. Inverse Different graph y=x; x y; solve for y X-axis: left right … terms of x Top – bottom *HINT* x=!!!!

8. Volume Methods

9. Disk Method Revolved around the x-axis Disk- “cylinders” r= the function value *always square the radius a.k.a the function value

10. Washer Method Y-axis in terms of y: Outer curve – inner curve dy or dx

11. Shell Method Rotate about the y-axis top curve -bottom curve In terms of x dy or/ dx

12. Area is connected to Rectangular approximation method because both involve finding the area under a curve or within a curve. How is Area linked to Rectangular Approximation Method?