find the area in the first quadrant bounded above by y=sinx, below by the x-axis, and to the right by x=pi/2 is divided into two equal parts by the line x=a. Find a.
To find the value of "a" that divides the area in the first quadrant into two equal parts, we need to perform the following steps:
Step 1: Visualize the region in the first quadrant bounded by y = sin(x), the x-axis, and x = π/2.
Step 2: Determine the total area of the region by integrating the function y = sin(x) from x = 0 to x = π/2.
Step 3: Set up an integral representing one-half of the total area and solve for a by integrating the function y = sin(x) from x = 0 to x = a.
Step 4: Solve the equation obtained in step 3 for the value of "a" that divides the area into two equal parts.
Let's go through each step in detail:
Step 1: Visualize the region in the first quadrant bounded by y = sin(x), the x-axis, and x = π/2.
The region we are interested in is the area beneath the curve y = sin(x) in the first quadrant, bounded by the x-axis from below and x = π/2 from the right. It will look like a triangular region.
Step 2: Determine the total area of the region.
To find the total area, we need to integrate the function y = sin(x) from x = 0 to x = π/2. The integral represents the signed area between the curve and the x-axis.
∫[0,π/2] sin(x) dx
Evaluating this integral will give us the total area of the region.
Step 3: Set up an integral representing one-half of the total area and solve for a.
We want to find the value of "a" that divides the total area into two equal parts. To do this, we need to set up an integral representing one-half of the total area and solve for "a".
Let's denote the half-area as A/2. So our integral becomes:
∫[0,a] sin(x) dx = A/2
Step 4: Solve the equation obtained in step 3 for the value of "a" that divides the area into two equal parts.
By integrating the function y = sin(x) from x = 0 to x = a and setting it equal to A/2, we can solve for "a". This will give us the value of "a" that divides the area into two equal parts.
∫[0,a] sin(x) dx = A/2
Evaluating this integral will give us the half-area. Setting it equal to A/2, we can solve for "a".
This method allows us to find the value of "a" that divides the area in the first quadrant into two equal parts.