Roughly sketch the region enclosed by the curves y = sin x, y = cos x and the x - axis
between x = 0 and x = p/ 2 . Also find the area of this region.
As you can see from the graphs at
http://www.wolframalpha.com/input/?i=plot+y+%3D+sin+x%2C+y+%3D+cos+x
the region is roughly triangular. The graphs intersect at x=π/4. Using vertical strips of width dx, you have the area
a = ∫[0,π/4] sinx dx + ∫[π/4,π/2] cosx dx
= -cosx [0,π/4] + sinx[π/4,π/2]
= -1/√2 +1 + 1 - 1/√2
= 2 - 2/√2
= 2 - √2
Note that if you had used horizontal strips of width dy, then
a = ∫[0,1/√2] arccos(y)-arcsin(y) dy
but that's a bit messy. However, wolframalpha confirms that you get the same result:
http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0%2C1%2F%E2%88%9A2%5D+%28arccos%28y%29-arcsin%28y%29%29+dy
To sketch the region enclosed by the curves y = sin x, y = cos x, and the x-axis between x = 0 and x = π/2, we can follow these steps:
1. Draw the coordinate axes (x-axis and y-axis) on a graph paper.
2. Mark the points (0, 0) and (π/2, 0) on the x-axis.
3. Plot the curve y = sin x by marking points for different values of x between 0 and π/2. Start with x = 0, which corresponds to y = sin(0) = 0. Then, plot a few more points by incrementing x, and connecting these points to form a smooth curve. This curve represents the function y = sin x.
4. Similarly, plot the curve y = cos x using the same method. Start with x = 0, which corresponds to y = cos(0) = 1. Plot a few more points by incrementing x, and connect them to form a smooth curve. This curve represents the function y = cos x.
5. Finally, shade the region enclosed by the curves y = sin x, y = cos x, and the x-axis between x = 0 and x = π/2. This shaded region represents the region enclosed by the given curves.
To find the area of this region, we can use the definite integral. The area between two curves y = f(x) and y = g(x) over an interval [a, b] is given by the integral:
Area = ∫[a, b] (f(x) - g(x)) dx
In this case, we need to find the area enclosed between y = sin x, y = cos x, and the x-axis over the interval [0, π/2]. So, we can calculate the area using the integral:
Area = ∫[0, π/2] (sin x - cos x) dx
To evaluate this integral, we can use integration techniques and find the antiderivative of (sin x - cos x) with respect to x. Then, we substitute the limits of integration and calculate the difference to find the area.