- Hour exam 2
- Tomorrow (Mar. 31)
- Material since 1st exam (e.g., multivariable functions, limits, partial derivatives, gradients, directional derivatives, tangents, extreme values, optimization, etc.)
- Rules and format similar to first exam, especially open-references and
calculators
- But remember to indicate where you use a calculator
- Section 15.1
- Like partial derivatives, but with integration
- Notably idea of treating most variables as constants while integrating with respect to another
- Divide region into rectangles to approximate volume
- Riemann sum over R
- Integral is limit as number of rectangles approaches ∞
- Integral is limit as number of rectangles approaches ∞
- To calculate: integrate wrt 1 variable, then wrt other
- Fubini’s Theorem: order of integration doesn’t matter
- Like partial derivatives, but with integration
- Examples
- Integrate f(x,y) = xy over 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1
- Why do you not see many indefinite multiple integrals? Because
“constant” terms are messy
- Why do you not see many indefinite multiple integrals? Because
“constant” terms are messy
- Integrate f(x,y,z) = xy2 + yz over -1 ≤ x ≤ 1, 0 ≤ y ≤ 2, 1
≤ z ≤ 2
- Evaluating these is easier with a tool such as muPad
- Evaluating these is easier with a tool such as muPad
- Integrate f(x,y) = xy over 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1
- (After exam)
- Double integrals over non-rectangular regions
- Read sections 15.2 and 15.3