course Mth 174
How would you do the folowing Riemann sum and integral inside x^2 + y^2 = 10 within 1st quadrant, using horizontal strip of width `dy. Would 10 be your r
x^2 + y^2 = 10 describes a circle of radius sqrt(10) about the origin.
A horizontal strip is characterized by its y coordinate, which we might call y_i, and its width, which we might call `dy_i.
At y coordinate y_i, we have x^2 + y_i ^ 2 = 10, so that x = +- sqrt(10 - y_i ^2). These are the x coordinates of the two points at which the horizontal line y = y_i intersects the circle.
The strip extends from (-sqrt(10 - y_i ^ 2) , y_i) to (sqrt(10 - y_i ^ 2), y_i), a horizontal distance of 2 sqrt(10 - y_i ^ 2). The width of the strip is `dy_i, so its area is
`dA_i = 2 sqrt(10 - y_i ^ 2) * `dy_i.
Summing up all such areas we get sum(`dA_i) = sum(2 sqrt(10 - y_i ^ 2) * `dy_i), which as the interval widths uniformly approach zero approaches the integral
integral(2 sqrt(10 - y^2) dy, y from -10 to 10).