Find the number a such that the line x = a divides the region bounded by the curves x = y^2 − 1 and the y-axis into 2 regions with equal area. Give your answer correct to 3 decimal places.
To find the number "a" such that the line x = a divides the region bounded by the curves x = y^2 − 1 and the y-axis into two regions with equal area, we need to first set up the integral to find the total area between the curves and then find the value of "a" that makes the two regions equal in area.
Let's start by finding the total area between the curves. The region between the curves and the y-axis is bounded by y = 0 and y = sqrt(x + 1), where sqrt denotes the square root function.
We can set up the integral as follows:
Total Area = ∫[a, b] (sqrt(x + 1) - 0) dx
To find the limits of integration, we need to solve the equation y = sqrt(x + 1) for x, when y = 0:
0 = sqrt(x + 1)
0 = x + 1
x = -1
So, the limits of integration are from a to -1 (since the region is bounded by x = -1 on the left and the line x = a on the right).
Now, we integrate to find the total area:
Total Area = ∫[a, -1] sqrt(x + 1) dx
To find the area of each region, we equate them to 1/2 of the total area:
1/2 Total Area = ∫[a, b/2] sqrt(x + 1) dx
where b/2 is the upper limit to divide the total area into two regions with equal areas.
Now, we set up the equation and solve for "a":
∫[a, -1] sqrt(x + 1) dx = 1/2 ∫[a, b/2] sqrt(x + 1) dx
Integrating both sides of the equation, we get:
[2/3 (x + 1)^(3/2)] evaluated from a to -1 = [1/3 (x + 1)^(3/2)] evaluated from a to b/2
Now, substitute the limits of integration and simplify the equation:
2/3 (-1 + 1)^(3/2) - 2/3 (a + 1)^(3/2) = 1/3 (b/2 + 1)^(3/2) - 1/3 (a + 1)^(3/2)
Simplifying this equation, we can cancel out some terms:
-2/3 (a + 1)^(3/2) = 1/3 (b/2 + 1)^(3/2) - 1/3 (a + 1)^(3/2)
Multiplying through by -3/2, we get:
a = (b/2 + 1)^(3/2) - (a + 1)^(3/2)
Now, we need to solve this equation to find the value of "a" such that the two regions have equal areas.
Unfortunately, solving this equation analytically might not be straightforward. However, we can use numerical methods or a graphing calculator to approximate the value of "a". By plotting the graphs of the curves and looking for the point where the two areas are equal, we can find the approximate value of "a" correct to 3 decimal places.