Question

Calculate the integral of f(x,y)=7x over the region D bounded above by y=x(2-x) and below by x=y(2- y).

Hint:Apply the quadratic formula to the lower boundary curve to solve for y as a function of x.

Answer 1

The student is asked to calculate the integral of a function over a bounded region by first applying the quadratic formula to the lower boundary and then setting up a double integral with the established limits.

Step-by-step explanation:

The student is tasked with finding the integral of the function f(x,y) = 7x over the region D bounded above by the curve y = x(2-x) and below by the curve rearranged from x = y(2-y) after applying the quadratic formula. The process involves finding the limits of integration for y in terms of x, then integrating the function with respect to x within these bounds.

To start, let's solve the lower boundary curve for y as a function of x. The equation x = y(2-y) is equivalent to y^2 - 2y + x = 0. Applying the quadratic formula, we get y = 1 ± √{1-x}. We are interested in the positive square root because it represents the lower boundary of D, which means y = 1 - √{1-x}.

Using these bounds, the integral of f(x,y) over D becomes a double integral which can be written as ∫∫_{D} 7x dy dx. We first integrate with respect to y, using the bounds from the equations of the curves, and then integrate with respect to x.

Answer 2

The integral of f(x,y) over region D bounded by two curves is calculated by first applying the quadratic formula to solve for y in terms of x, determining the integral bounds, and then integrating 7x over the region defined by these bounds.

Step-by-step explanation:

To calculate the integral of f(x,y) = 7x over the region D, we must first understand the boundary curves that define this region. The region D is bounded above by y = x(2 - x) and below by x = y(2 - y). The curve x = y(2 - y) needs to be rearranged to solve for y as a function of x, which requires the use of the quadratic formula.

The quadratic equation can be rewritten as y^2 - 2y + x = 0. Applying the quadratic formula y = (-b ± √(b^2 - 4ac))/(2a), we find the values of y in terms of x. The boundaries for x can be determined by setting y to zero in the equation y = x(2 - x), which yields x = 0 and x = 2.

With the bounds of x and the quadratic solutions for y, the double integral over the region D can be set up and calculated. Assuming that both functions y = x(2 - x) and y = (-b + √(b^2 - 4ac))/(2a) describe the same geometric region, we can integrate f(x, y) over the region by integrating 7x dy for y from the lower boundary to the upper boundary and then integrating the result dx from 0 to 2.