find the area of the region bounded by the curves y=x^2-1 and y =cos(x)
the hardest part is finding the intersection of the two curves,
x^2 - 1 = cosx
I will let Wolfram do it:
http://www.wolframalpha.com/input/?i=x%5E2+-+1+%3D+cosx
Because of the symmetry we could go from 0 to 1.1765 and double the answer
area = 2 ∫(cosx - x^2 + 1)dx from 0 to 1.1765
= 2[sinx - (1/3)x^3 + x] from 0 to 1.1765
= 2(.923237... - .542818... + 1.1765 - (0 - 0 + 0))
= appr 5.285
To find the area of the region bounded by the curves y = x^2 - 1 and y = cos(x), you need to find the points of intersection where these two curves intersect each other. The area can be calculated by integrating the difference in the y-values of the curves.
Step 1: Finding the points of intersection:
Set the two equations equal to each other and solve for x:
x^2 - 1 = cos(x)
Unfortunately, this equation cannot be solved algebraically as it involves both a polynomial and a trigonometric function. Consequently, we will need to use numerical methods or graphical approaches to approximate the points of intersection.
Step 2: Numerical Method (Approximation):
One numerical method for approximating the points of intersection is the bisection method. Here's a step-by-step explanation of how to use this method:
1. Start by graphing the two curves y = x^2 - 1 and y = cos(x). You can use software like Desmos or graphing calculators to visualize the curves.
2. Observe the graph to estimate the approximate x-values where the curves intersect. In this case, it appears that the curves intersect at around x ≈ -1.5, x ≈ -0.5, x ≈ 0.5, and x ≈ 1.5. These are only rough estimations; the actual values may differ.
3. Use the bisection method to refine your estimates. The bisection method involves repeatedly taking the midpoint between two x-values, checking if the corresponding y-values have opposite signs, and then narrowing down the interval where the curve intersects.
4. Start with an interval, [a, b], where you believe the curves intersect. Choose two points, x1 and x2, that are reasonably close to each other within this interval, and calculate their corresponding y-values, y1 and y2. Ensure that y1 and y2 have opposite signs.
5. Find the midpoint, c, between x1 and x2: c = (x1 + x2) / 2.
6. Calculate the corresponding y-value, y_c, for the midpoint c.
7. Repeat steps 5 and 6 until you find a midpoint with a y-value close to zero.
8. Once you have determined the approximate x-values of the points of intersection, proceed to the next step.
Step 3: Calculating the area using integration:
Now that we have found the approximate x-values of the points of intersection, we can proceed to calculate the area bounded by the curves. Recall that the area between two curves can be calculated by integrating the difference in their y-values.
1. Determine the integral limits by identifying the x-values where the curves intersect. Let's call these x-values x1, x2, x3, and x4. The limits of integration for x are x1 to x2 and x3 to x4.
2. Calculate the difference in y-values for each segment.
3. Set up the integral for each segment by subtracting the y-values of the lower curve from the upper curve.
4. Integrate the resulting expressions separately for each segment.
5. Finally, subtract the integral values obtained for each segment to find the total area.
Please note that the steps provided here are a general guide, and the exact calculations and methods might vary depending on the specific problem at hand.