If the region in the first quadrant bounded by the graph of y=cosx and the interval[0,pi/2] is bisected by the line x=c, guess c.
You need the
integral [cosx]dx from0 to c = integral [cosx]dx from c to π/2
sinc = sin π/2 - sinc
2sinc = sin π/2
sinc = 1/2
c = π/6
Why are you supposed to "guess" ?
thats what the question said
To find the value of c that bisects the region in the first quadrant bounded by the graph of y = cos(x) and the interval [0, pi/2], we can follow these steps:
Step 1: Understand the problem.
- We need to find the x-coordinate of the point where the line x = c bisects the bounded region.
Step 2: Visualize the problem.
- Draw the graph of y = cos(x) in the first quadrant, emphasizing the interval [0, pi/2].
- Also, draw a vertical line x = c, where c is the value we are trying to find.
Step 3: Analyze the situation.
- Notice that the graph of y = cos(x) is decreasing from x = 0 to x = pi/2 in the first quadrant.
- We want to find the x-coordinate c, which divides the bounded region equally.
Step 4: Apply the bisection concept.
- Since we want the region to be divided equally, the area on the left side of the line x = c should be the same as the area on the right side of the line x = c.
Step 5: Use symmetry to find c.
- By observing the symmetry of the graph of y = cos(x) with respect to the y-axis, we can conclude that the line x = c will pass through the midpoint of the interval [0, pi/2].
- Hence, c = (0 + pi/2) / 2 = pi/4.
So, the value of c that bisects the region in the first quadrant is pi/4.