graph y=cos(pi*x/2) and y=1-x^2 and use integration to find the area in between the curve.

Okay, so when I graph these two I see that they like overlap during the [-1,1] x interval. But maybe there is still a small gap in between? But I'm not sure if the [-1,1] x interval is correct. Please help!

Thanks in advance

I see what you mean, the two curves appear to coincide form -1 to 1

http://www.wolframalpha.com/input/?i=y%3Dcos(pi*x%2F2),+y%3D1-x%5E2

Because of the symmetry, I looked closer from 0 to 1 and it shows that the parabolo lies above the cosine curve for that domain, and of course must do the same for x from -1 to 0
http://www.wolframalpha.com/input/?i=y%3Dcos(pi*x%2F2),+y%3D1-x%5E2+,+for+0+to+1

Also notice that they meet at x = 0 , at (0,1)

So we will just take the area from 0 to 1 and double it

area - 2∫(1 - x^2 - cos(πx/2) dx from 0 to 1

take over, the integration is straight-forward.

It's not surprising that the two curves are so close. As you learn about functions, you will come across the Taylor Series. Any function can be approximated over a limited domain by a polynomial. This is handy, because polynomials are well behaved.

Anyway, you will find that

cos(u) = 1 - u^2/2! + u^4/4! - u^6/6! ...

Note how both functions are even functions. And the difference between cos(u) and 1-u^2/2! is very tiny -- 4th and higher powers. On the domain (-1,1) those higher powers of fractions are quite small.

Plug in u = π/2 x and you have

cos(π/2 x) ≈ 1 - (π/2)^2/2! x^2

On the interval [-1,1] that is just 1- π^2/8 x^2. Since π^2/8 > 1, that parabola lies just below 1-x^2. as seen here in the restricted graph:

http://www.wolframalpha.com/input/?i=y%3Dcos%28pi*x%2F2%29,+y%3D1-x^2+for+-1%3C%3Dx%3C%3D1

To find the area between the curves y = cos(pi*x/2) and y = 1 - x^2, we need to first determine the points of intersection between the two curves.

Setting the equations equal to each other:
cos(pi*x/2) = 1 - x^2

Rearranging:
x^2 + cos(pi*x/2) - 1 = 0

Since it is not possible to solve this equation analytically, we can use numerical methods or a graphing calculator to approximate the values of x at the points of intersection.

Using a graphing calculator, we find that the points of intersection occur approximately at x ≈ -0.77 and x ≈ 0.77.

Now, to find the area between the curves, we can set up an integral:

Area = ∫[a, b] (f(x) - g(x)) dx

where a and b are the x-coordinates of the points of intersection, and f(x) and g(x) are the upper and lower curves, respectively.

In this case, the upper curve is y = cos(pi*x/2), and the lower curve is y = 1 - x^2.

Thus, the integral setup would be:

Area = ∫[-0.77, 0.77] (cos(pi*x/2) - (1 - x^2)) dx

To compute the integral, we can use any appropriate integration method like numerical integration (e.g., Simpson's Rule) or a graphing calculator.

To find the area between the curves y = cos(pi*x/2) and y = 1 - x^2, you'll first need to determine the points where the curves intersect. This will give you the limits of integration for calculating the area.

Let's start by setting the two equations equal to each other:

cos(pi*x/2) = 1 - x^2

Now, we can rearrange the equation to isolate x:

x^2 + cos(pi*x/2) - 1 = 0

To solve this equation for x, we can use numerical methods or graphing calculators. In this case, we can use a graphing calculator to find the approximate values of x, where the curves intersect on the interval [-1, 1].

Graphically, we observe that the curves intersect at x ≈ -0.732 and x ≈ 0.732.

Now that we have the limits of integration, we can find the area between the curves by taking the definite integral of the positive difference between the curves over the given interval:

Area = ∫[a, b] (f(x) - g(x)) dx,

where f(x) is the curve on top (in this case, f(x) = cos(pi*x/2)), g(x) is the curve on the bottom (in this case, g(x) = 1 - x^2), and [a, b] represents the interval of integration.

Therefore, the area between the curves on the interval [-1, 1] can be calculated as:

A = ∫[-0.732, 0.732] (cos(pi*x/2) - (1 - x^2)) dx

You can use integration techniques, such as substitution or integration by parts, to evaluate this integral and find the exact value of the area. However, if you prefer a numerical approximation, you can use a tool like a graphing calculator or software to calculate the definite integral.