Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order. ∬_(D)ydA,quad D is boundad by y=x-20,x=y^(2)

Question

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order.  ∬_(D)ydA,quad D is boundad by  y=x-20,x=y^(2)

Bytelearn · Accepted Answer

Understand Region of Integration: Understand the region of integration D. The region D is bounded by the curves y = x - 20 and x = y^2. To find the points of intersection, we set x - 20 = y and y^2 = x equal to each other and solve for y. y = y^2 - 20 y^2 - y - 20 = 0 This is a quadratic equation in y.Solve Quadratic Equation: Solve the quadratic equation to find the points of intersection. We can factor the quadratic equation as follows: (y - 5)(y + 4) = 0 This gives us two solutions for y: y = 5 and y = -4. These are the y-values of the points of intersection. To find the corresponding x-values, we substitute these y-values back into either of the original equations. Let's use x = y^2. For y = 5, x = 5^2 = 25. For y = -4, x = (-4)^2 = 16. So the points of intersection are (25, 5) and (16, -4).Set Up Iterated Integral (dy dx): Set up the iterated integral for the order dy dx. To integrate with respect to y first (dy) and then x (dx), we need to find the limits of integration for y in terms of x. From the equations of the curves, we have y = x - 20 and y = sqrt(x). The limits of integration for y are from the lower curve (x - 20) to the upper curve (sqrt(x)). The limits of integration for x are from the leftmost point of intersection (x = 16) to the rightmost point of intersection (x = 25). The iterated integral is: ∫ from x=16 to x=25 ∫ from y=x-20 to y=sqrt(x) y dy dxSet Up Iterated Integral (dx dy): Set up the iterated integral for the order dx dy. To integrate with respect to x first (dx) and then y (dy), we need to find the limits of integration for x in terms of y. From the equations of the curves, we have x = y^2 and x = y + 20. The limits of integration for x are from the left curve (y^2) to the right curve (y + 20). The limits of integration for y are from the bottommost point of intersection (y = -4) to the topmost point of intersection (y = 5). The iterated integral is: ∫ from y=-4 to y=5 ∫ from x=y^2 to x=y+20 y dx dyDetermine Easier Order: Determine the easier order of integration. To determine which order of integration is easier, we should consider the complexity of the functions involved. The order dx dy involves integrating a constant function with respect to x, which is simpler than integrating a linear function with respect to y. Therefore, the easier order of integration is dx dy.Evaluate Double Integral: Evaluate the double integral using the easier order dx dy. We will now evaluate the integral: ∫ from y=-4 to y=5 ∫ from x=y^2 to x=y+20 y dx dy First, integrate with respect to x: ∫ from y=-4 to y=5 [yx] from x=y^2 to x=y+20 dy = ∫ from y=-4 to y=5 [y(y+20) - y(y^2)] dy = ∫ from y=-4 to y=5 [20y - y^3] dy Now, integrate with respect to y: ∫ from y=-4 to y=5 [20y - y^3] dy = [10y^2 - (1/4)y^4] from y=-4 to y=5 = [10(5)^2 - (1/4)(5)^4] - [10(-4)^2 - (1/4)(-4)^4] = [250 - 625/4] - [160 - 256/4] = [250 - 156.25] - [160 - 64] = [93.75] - [96] = -2.25Verify and Conclude: Verify the result and conclude the solution. The final value of the double integral is -2.25. This is the result after evaluating the integral using the easier order of integration, which was determined to be dx dy.

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order. $\iint_{D} y d A, \quad D$ is boundad by $y=x-20, x=y^{2}$

Full solution

More problems from Double integration by reversing the order

Related topics

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order. ∬DydA,D \iint_{D} y d A, \quad D ∬D​ydA,D is boundad by y=x−20,x=y2 y=x-20, x=y^{2} y=x−20,x=y2

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order. $\iint_{D} y d A, \quad D$ is boundad by $y=x-20, x=y^{2}$