Find the area common to the circles r=a cos(theta) and r= a sin(theta) by double integration
Answers
Step-by-step explanation:
Polar Rectangular Regions of Integration
When we defined the double integral for a continuous function in rectangular coordinates—say, g over a region R in the xy-plane—we divided R into subrectangles with sides parallel to the coordinate axes. These sides have either constant x-values and/or constant y-values. In polar coordinates, the shape we work with is a polar rectangle, whose sides have constant r-values and/or constant θ-values. This means we can describe a polar rectangle as in Figure 5.28(a), with R={(r,θ)|a≤r≤b,α≤θ≤β}.
In this section, we are looking to integrate over polar rectangles. Consider a function f(r,θ) over a polar rectangle R. We divide the interval [a,b] into m subintervals [ri−1,ri] of length Δr=(b−a)/m and divide the interval [α,β] into n subintervals [θj−1,θj] of width Δθ=(β−α)/n. This means that the circles r=ri and rays θ=θj for 1≤i≤m and 1≤j≤n divide the polar rectangle R into smaller polar subrectangles Rij (Figure 5.28(b)).