Calculus Review Topics

Area Between Curves
This topic is difficult because I often forget to take the anti-derivative of the functions before solving for the area.

Example:  Find area of the region bounded by: f(x) = 2x - 2 and g(x) = -x² - 2x + 3

Find area of the region bounded by: f(x) = 2x³ - 1 and g(x) = 2x - 1

Find area of the region bounded by f(x) = x and g(x) = √x

Volume of a Region Revolved Around an Axis Using Disks
This topic can be difficult when you forget which is the width and which is the height. Usually this is just a foolish mistake, however.

Example: Using the region bounded by f(x) = √x and x = 4, find the volume revolving about the x-axis.

Using the region bounded by f(x) = 2x and x=5, find the volume revolving about the x-axis.

Volume of a Region Revolved About an Axis Using Washers

For me, this topic is usually hard as a result of changing the width and height.

Example: Find the volume of area bounded by x = 0, y = x + 4 and y = (1/2)x² revolved around x = -1.

Find the volume of the solid obtained by rotating the portion of the region bounded by y = x^(1/3), x = 0 and y = x / 4.

Volume of a Region evolved About an Axis Using Cylindrical Shells
As a result of spending a lot of time on washers and disks, one hard part is remembering to switch which values we use based on what the region is being revolved around.

Example Find the volume of region bounded by y = 2x³, y = 2x, x =1, and x=2; revolved about the y-axis.

Find the volume of region bounded by y = 5x², x = 1 and y = 1; revolved about the y-axis.

Applying the Fundamental Theorem of Calculus Parts 1 and 2

Sometimes problems can arise when you forget to multiply by the derivative if you are not dealing solely with a variable when using the FTC part 1. For the FTC part 2, sometimes you might forget to use the anti-derivative to evaluate integrals.

Example: Evaluate f'(x) when f(x) = ∫_-1^4x (4t - 10t²)dt.

Evaluate f'(x) when f(x) = ∫_x^{3x^2} cos(θ)dθ

Evaluate the following definite integral: f(x) = ∫₀¹x³dx