Question

Set up the integral(s) that give the area of the region bounded by the given equations. Show the equivalent set up with respect to x as well as with respect to y. 9.y=x,y=8x2,y=2 with respect to x with rerpect to y

Answer 1

To calculate the area between y = x, y = 8x², and y = 2, set up definite integrals with either x or y as the variable of integration, based on the intersection points of the curves and applying the functions appropriately.

Step-by-step explanation:

To set up the integral(s) for the area of the region bounded by the equations y = x, y = 8x2, and y = 2 with respect to x, we first identify the intersection points of the curves. Next, we set up the integral of the top function minus the bottom function, evaluated from the leftmost to the rightmost intersection point. Similarly, to set up the integral with respect to y, we first solve each equation for x in terms of y and then set up the integral from the lowest y-value to the highest y-value of the vertical strip, calculating the rightward function minus the leftward function.

With respect to x:

The integrals for the area are:

∫ (1 - 8x2) dx from x1 to x2

With respect to y:

The integrals for the area are:

∫ (√(y/8) - y/2) dy from y1 to y2