Find the area of the shaded region below.
x=(y^2)-2
x=e^y
y=1
y=-1
I was doing a horizontal split so I had the integral from -1 to 1 (y^2-2)-e^y dy and had the answer -5.683... and it's wrong. What am I doing wrong?
Update: Oops, nevermind; accidentally switched the 2 functions. Should be 5.683. Thanks anyways!
To find the area of the shaded region, you need to integrate the absolute difference between the two curves as they intersect. Here's how you can approach this:
1. Plot the graphs of the equations x = y^2 - 2 and x = e^y on a coordinate plane.
2. Determine the points of intersection between the two curves. To do this, set the two equations equal to each other:
y^2 - 2 = e^y
You can solve this equation numerically or graphically to find the approximate values.
3. Once you have the points of intersection, calculate the area under the curves between these points. Since the shaded region lies between y = -1 and y = 1, integrate the absolute difference function:
A = ∫[y=-1 to 1] |(y^2 - 2) - e^y| dy
Note that the absolute value is necessary because the curves may intersect in a way that their difference becomes negative at some points.
4. Evaluate the integral to find the area of the shaded region.
It seems like you made an error in your integration. Make sure you take the absolute difference of the curves and evaluate the integral correctly.
To find the area of the shaded region, you need to first determine the points at which the curves intersect. From the given equations, we have three curves: x = (y^2) - 2, x = e^y, and y = 1.
Let's find the points of intersection:
1. Setting x = e^y and x = (y^2) - 2 equal to each other:
e^y = (y^2) - 2
This equation is transcendental and cannot be easily solved algebraically. So, we will need to solve it numerically. One way to do this is to graph both functions and find their points of intersection. Alternatively, you can use a numerical method like the Newton-Raphson method to approximate the solutions.
2. Setting y = 1 in x = (y^2) - 2:
x = (1^2) - 2
x = -1
So, we have one point of intersection at (x, y) = (-1, 1).
Now, the area of the shaded region can be found by evaluating the appropriate integral. Since you mentioned that you were using a horizontal split, it seems like you were on the right track. However, your integral seems to be set up incorrectly.
To find the area between two curves using integration, you need to subtract the functions and integrate over the appropriate interval. In this case, the shaded region is bounded by the curves x = (y^2) - 2 and x = e^y between the values y = -1 and y = 1.
Therefore, the correct integral to find the area is:
A = ∫[y=-1 to y=1] [(y^2) - 2 - e^y] dy
Evaluating this integral should give you the correct area of the shaded region. If you encounter any difficulties in evaluating the integral, you can use numerical methods or integral calculators available online.