Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.x = 1 − y2, x = y2 − 1The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the second curve, changes direction at the point (0, 0.5), goes down and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the second curve, and exits the window in the fourth quadrant.The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the first curve, changes direction at the point (0, −0.5), goes up and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the first curve, and exits the window in the first quadrant.The region is below the first curve and above the second curve.The approximating rectangle occurs at x = 0.5, extends vertically from the second curve to the first curve, and has a width of Δx. The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at x = −1 crossing the second curve, changes direction at the point (0, 1), goes down and right becoming more steep, crosses the x-axis at x = 1 crossing the second curve, and exits the window in the fourth quadrant.The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at x = −1 crossing the first curve, changes direction at the point (0, −1), goes up and right becoming more steep, crosses the x-axis at x = 1 crossing the first curve, and exits the window in the first quadrant.The region is below the first curve and above the second curve.The approximating rectangle occurs at x = 0.5, extends vertically from the second curve to the first curve, and has a width of Δx. The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming more steep, crosses the y-axis at y = −1 crossing the second curve, changes direction at the point (1, 0), goes up and left becoming less steep, crosses the y-axis at y = 1 crossing the second curve, and exits the window in the second quadrant.The second curve enters the window in the fourth quadrant, goes up and left becoming more steep, crosses the y-axis at y = −1 crossing the first curve, changes direction at the point (−1, 0), goes up and right becoming less steep, crosses the y-axis at y = 1 crossing the first curve, and exits the window in the first quadrant.The region is left of the first curve and right of the second curve.The approximating rectangle occurs at y = 0.5, extends horizontally from the second curve to the first curve, and has a height of Δy. The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming more steep, crosses the y-axis at approximately y = −0.71 crossing the second curve, changes direction at the point (0.5, 0), goes up and left becoming less steep, crosses the y-axis at approximately y = 0.71 crossing the second curve, and exits the window in the second quadrant.The second curve enters the window in the fourth quadrant, goes up and left becoming more steep, crosses the y-axis at approximately y = −0.71 crossing the first curve, changes direction at the point (−0.5, 0), goes up and right becoming less steep, crosses the y-axis at approximately y = 0.71 crossing the first curve, and exits the window in the first quadrant.The region is left of the first curve and right of the second curve.The approximating rectangle occurs at y = 0.5, extends horizontally from the second curve to the first curve, and has a height of Δy.Find the area of the region.

Question

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.x = 1 − y2,   x = y2 − 1The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the second curve, changes direction at the point (0, 0.5), goes down and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the second curve, and exits the window in the fourth quadrant.The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the first curve, changes direction at the point (0, −0.5), goes up and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the first curve, and exits the window in the first quadrant.The region is below the first curve and above the second curve.The approximating rectangle occurs at x = 0.5, extends vertically from the second curve to the first curve, and has a width of Δx. The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at x = −1 crossing the second curve, changes direction at the point (0, 1), goes down and right becoming more steep, crosses the x-axis at x = 1 crossing the second curve, and exits the window in the fourth quadrant.The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at x = −1 crossing the first curve, changes direction at the point (0, −1), goes up and right becoming more steep, crosses the x-axis at x = 1 crossing the first curve, and exits the window in the first quadrant.The region is below the first curve and above the second curve.The approximating rectangle occurs at x = 0.5, extends vertically from the second curve to the first curve, and has a width of Δx. The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming more steep, crosses the y-axis at y = −1 crossing the second curve, changes direction at the point (1, 0), goes up and left becoming less steep, crosses the y-axis at y = 1 crossing the second curve, and exits the window in the second quadrant.The second curve enters the window in the fourth quadrant, goes up and left becoming more steep, crosses the y-axis at y = −1 crossing the first curve, changes direction at the point (−1, 0), goes up and right becoming less steep, crosses the y-axis at y = 1 crossing the first curve, and exits the window in the first quadrant.The region is left of the first curve and right of the second curve.The approximating rectangle occurs at y = 0.5, extends horizontally from the second curve to the first curve, and has a height of Δy. The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph.The first curve enters the window in the third quadrant, goes up and right becoming more steep, crosses the y-axis at approximately y = −0.71 crossing the second curve, changes direction at the point (0.5, 0), goes up and left becoming less steep, crosses the y-axis at approximately y = 0.71 crossing the second curve, and exits the window in the second quadrant.The second curve enters the window in the fourth quadrant, goes up and left becoming more steep, crosses the y-axis at approximately y = −0.71 crossing the first curve, changes direction at the point (−0.5, 0), goes up and right becoming less steep, crosses the y-axis at approximately y = 0.71 crossing the first curve, and exits the window in the first quadrant.The region is left of the first curve and right of the second curve.The approximating rectangle occurs at y = 0.5, extends horizontally from the second curve to the first curve, and has a height of Δy.Find the area of the region.

Accepted Answer

The region is bounded by the curves x=1-y2, x=y2-1.
We have sketched the graph of the region bounded…