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UM - MATH 116 - Study Guide - Final

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Math 116 Exam 1 Study GuideWe also discuss several other topics like Giave a sample of format in writing a letter.

- Fundamental Theorem of Calculus I

- If f is continuous on [a, b] and F’(x) = f(x) then

- Theorems about definite integrals

- If f is continuous everywhere and a, b, and c are any number (not necessarily a < b < c), then…

- 1.
- 2.

- Linearity Theorem

- If f and g are continuous on [a, b] and c is equal to any real number, then…

- 1.
- 2.

- Fundamental Theorem of Calculus II

- If f is continuous on [a, b ] and F is an antiderivative of f: F’(x) = f(x), then

- U - substitution

- If an integral is in the form , you can put u = g(x), du = g’(x) dx, and modify the bounds appropriately: .

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- Integration by parts

- ← explicit
- ← common form
- For indefinite integration…

- Picking u and dv:

- simpler than u
- easy to integrate

- General priority of picking u (not always):

- Log
- Inverse trigonometry
- Polynomial
- Exponential
- Trigonometry

- “Lipet” Rule

- Trapezoidal Rule=

- Equivalent to getting a secant line

- Secant line above the graph → overestimate concave up

- Secant line below the graph → underestimate concave down

- Midpoint Rule

- Equivalent to a tangent line

- Tangent line below the graph → underestimate concave up

- Tangent line above graph- overestimate concave down

- Area and Volume

- Area under the curve: just the integral of the function
- Area between two curves: one we call the ceiling, the other is the floor

- Area =

- Volume of a 3D region

- Vertical strip revolving around a horizontal axis → washer method
- Vertical strip revolving around a vertical axis → shell method
- Horizontal strip revolving around a vertical axis → washer method
- Horizontal strip revolving around a horizontal axis → shell method

When given the freedom to slice, consider:

- How the functions are given y = f(x), x = f(y)
- Which variable the density function depends on

- Objects with prescribed cross sections perpendicular to a given axis: find the area of cross section in terms of the independent variable, accumulate or integrate them.

- Arc Length

- A graph y = f(x) between x = a and x = b has the length:

- Center of Mass

- x / y / z coordinate of center of mass
- ; where 𝓵(x) is the object’s density

- Mass with constant density:

- 3D Objects: mass = volume * density
- 2D Objects: mass = area * density

- Mass with non-constant density

- Divide the regions into small pieces in such a way that the density is approximately constant on each piece, then add together using an integral