How do I determine the area of a region bounded by a function and the x-axis if given y=cosx and interval is [0,ð/2]. This has to do with the Fundamental Theorem of Calculus. Thanks in advance!
FTC says that the area is the definite integral.
area = Int(cosx)[0,pi/2]
= sinx [0,pi/2]
= 1 - 0
= 1
To determine the area of a region bounded by a function and the x-axis, you can use the Fundamental Theorem of Calculus. Here's a step-by-step explanation of how to do it for the given function y = cos(x) and interval [0, π/2]:
1. First, let's sketch the graph of the function y = cos(x) in the given interval [0, π/2]. This will help us visualize the region we're trying to find the area of.
2. Next, we'll find the antiderivative (or the indefinite integral) of cos(x). The antiderivative of cos(x) is sin(x), so ∫cos(x) dx = sin(x) + C, where C is the constant of integration.
3. Now, we need to evaluate the definite integral of cos(x) over the interval [0, π/2]. The definite integral is ∫[0, π/2] cos(x) dx.
4. Using the Fundamental Theorem of Calculus, we can calculate the definite integral of cos(x) over [0, π/2] by subtracting the value of the antiderivative at the upper limit (π/2) from the value at the lower limit (0). So, the area is given by the formula A = ∫[0, π/2] cos(x) dx = sin(π/2) - sin(0).
5. Simplifying the expression, we find A = 1 - 0 = 1.
Therefore, the area of the region bounded by the function y = cos(x) and the x-axis over the interval [0, π/2] is 1.