Final answer:
To find the area of the region between the curves x = y² - 7 and x = 1 - y², we need to find their points of intersection. By setting the two equations equal to each other and solving, we find the points of intersection at (-3, 2) and (-3, -2). The area between the curves can then be found by calculating the definite integral of the difference between the two functions.
Step-by-step explanation:
To find the area of the region lying to the right of x = y² - 7 and to the left of x = 1 - y², we first need to find the points of intersection of these two curves. Setting x = y² - 7 equal to x = 1 - y², we get y² - 7 = 1 - y². Simplifying, we have 2y² = 8, which gives us y² = 4 and y = ±2. Plugging these values of y back into the equations, we can find the corresponding values of x. Substituting y = 2 into x = y² - 7, we get x = 2² - 7 = -3. Substituting y = -2 into x = y² - 7, we get x = (-2)² - 7 = -3. Therefore, the two curves intersect at the points (-3, 2) and (-3, -2).
To find the area between the curves, we need to calculate the definite integral of the difference between the two functions from x = -3 to the x-coordinate of the rightmost intersection point. In this case, the rightmost intersection point is (-3, 2), so we need to integrate the function (1 - y²) - (y² - 7) from x = -3 to x = -3. The integral of this function will give us the area of the region bounded by the two curves. Evaluating the integral, we get ∫[(1 - y²) - (y² - 7)] dx = ∫-2y² + 8 dx = -2∫y² dx + 8∫1 dx = -2(y³/3) + 8x + C. We can plug in the values x = -3 and x = -3 into this expression to find the area of the region between the two curves.