Multivariable Calculus (Calc III) - Complete Semester Course - Multivariable Calculus: Definition of Double Riemann integration

We look at the definition of the double Riemann integral of a continuous function z=f(x,y) over a rectangle R=[a,b]x[c,d]. We do one example with a double Riemann sum. (Unit 4 Lecture 1) In greater detail, we start by defining a rectangular domain 𝑅 in the xy-plane, denoted as [𝑎,𝑏] × [𝑐,𝑑], where 𝑥 varies from 𝑎 to 𝑏 and 𝑦 from 𝑐 to 𝑑. We consider a continuous function 𝑧=𝑓(𝑥,𝑦) defined over this rectangle. To approximate the volume under the surface 𝑧=𝑓(𝑥,𝑦), we partition 𝑅 into sub-rectangles and choose a representative point (𝑥_𝑖 , 𝑦_𝑗) for each sub-rectangle. The volume of each rectangular prism formed over the sub-rectangles is approximated as 𝑓(𝑥_𝑖,𝑦_𝑗) × Δ𝐴𝑖𝑗, where Δ𝐴𝑖𝑗 is the area of the sub-rectangular base. The double integral is computed by setting up and evaluating the Riemann sum. We sum over all sub-rectangles and then take the limit as the number of sub-rectangles approaches infinity. This process is demonstrated through a detailed example where we integrate the function 𝑓(𝑥,𝑦)=𝑥+𝑦 over a specific rectangular domain. The steps involve setting up a double Riemann sum and then taking the limit as the number of subdivisions approaches infinity. #calculus #multivariablecalculus #mathematics #doubleintegrals #integration #iitjammathematics #calculus3