Multivariable Calculus (Calc III) - Complete Semester Course - Double integration over general regions, Multivariable Calculus Unit 4 Lecture 3

We begin our lecture with an examination of double Riemann integrals over general regions, broadening our scope from rectangular domains. We introduce three types of regions: 1. Type 1 Regions: Defined by 𝑥 values between constants 𝑎 and 𝑏, and 𝑦 values moving between two curves, 𝑦=𝑔(𝑥) and 𝑦=ℎ(𝑥). The integral will take the form ∫∫𝑓(𝑥,𝑦) 𝑑𝑥 𝑑𝑦, where the outermost bounds are a to b, and the innermost are g(x) to h(x). 2. Type 2 Regions: Here, the roles of 𝑥 and 𝑦 are reversed. 𝑦 values are between constants 𝑐 and 𝑑, and 𝑥 values move between two curves, 𝑥=𝑚(𝑦) and 𝑥=𝑛(𝑦). The integral will take the form ∫∫𝑓(𝑥,𝑦) 𝑑𝑦 𝑑𝑥, where the outermost bounds are c to d, and the innermost are m(y) to n(y). 3. Type 3 Regions: These regions can be interpreted as either Type 1 or Type 2, offering flexibility in setting up integrals. Important remark: outermost bounds must always be constants (no variable expressions). Examples: 1. We compute a double integral over a Type 1 region defined by 0 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑦 ≤ 𝑥^3. This involves evaluating the inner integral with respect to 𝑦 and then the outer integral with respect to 𝑥. 2. We sketch and integrate over a domain 𝐷 with bounds 𝑦 ≤ 𝑥 ≤ 2𝑦+1 and −1 ≤ 𝑦 ≤ 3. 3. We do an example where we reverse the bounds of integration. 4. We demonstrate setting up a double integral representing the volume of a solid bounded by a cylinder and various planes. This involves identifying the bounds of integration and selecting the appropriate integration order (either 𝑑𝑦𝑑𝑥 or 𝑑𝑥𝑑𝑦). Properties of Double Integration * Linearity The integral of a linear combination of functions is equivalent to the sum of the integrals of each function, scaled appropriately. Mathematically: ∬ (𝑐⋅𝑓+𝑑⋅𝑔) 𝑑𝐴 = 𝑐⋅∬ 𝑓 𝑑𝐴 + 𝑑⋅∬ 𝑔 𝑑𝐴 * Monotonicity If 𝑓 ≤ 𝑔 over 𝐷, then their integrals follow the same inequality: ∬𝑓𝑑𝐴 ≤ ∬ 𝑔𝑑𝐴 * Integration Over Union of Regions The integral over the union of disjoint regions 𝐷1 and 𝐷2 is the sum of the integrals over each region: ∬_{𝐷1∪𝐷2} 𝑓 𝑑𝐴 = ∬_{𝐷1} 𝑓 𝑑𝐴 + ∬_{𝐷2} 𝑓 𝑑𝐴 * Integration Over Subsets If 𝑆⊆𝐷 (and f is nonnegative), then ∬𝑆𝑓𝑑𝐴 ≤ ∬𝐷𝑓𝑑𝐴. Factorization For a function 𝑓(𝑥,𝑦)=𝑔(𝑥)ℎ(𝑦) over a rectangle, the integral can be split: ∬ 𝑔(𝑥)ℎ(𝑦) 𝑑𝐴 = ∫𝑔(𝑥)𝑑𝑥 ⋅ ∫ℎ(𝑦)𝑑𝑦, with appropriate bounds for each single integral. #Calculus #Mathematics #RiemannIntegral #MultivariableCalculus #MathTutorial #CalculusLecture #MathEducation #IntegralCalculus #iitjammathematics #calculus3