Use left endpoints and 9 rectangles to find the approximation of the area of the region between the graph of the function cos2x and the x-axis over the interval [0, pi/2] . Round your answer to four decimal places.
since the graph in the interval is symmetric about (0,pi/4) the area is zero.
cos(2x) goes from 1 to -1 over the interval.
However, using the specified rectangles, the approximation is 0.1745
thank you!
To approximate the area of the region between the graph of the function cos(2x) and the x-axis using left endpoints and 9 rectangles, follow these steps:
Step 1: Divide the interval [0, π/2] into n subintervals.
Since we are using 9 rectangles, n = 9.
Step 2: Calculate the width of each subinterval.
The width (Δx) of each subinterval is given by Δx = (b - a) / n, where a is the lower limit of the interval (0 in this case) and b is the upper limit of the interval (π/2 in this case).
Δx = (π/2 - 0) / 9
= π/18
Step 3: Determine the endpoints for each subinterval.
Since we are using left endpoints, the left endpoint for each subinterval will be the starting point of that subinterval.
Subinterval 1: [0, π/18]
Subinterval 2: [π/18, 2π/18]
Subinterval 3: [2π/18, 3π/18]
...
Subinterval 9: [8π/18, π/2]
Step 4: Calculate the function value at each left endpoint.
Evaluate the function cos(2x) at each left endpoint of the subintervals.
For example:
At x = 0 (left endpoint of subinterval 1):
f(0) = cos(2 * 0) = cos(0) = 1
At x = π/18 (left endpoint of subinterval 2):
f(π/18) = cos(2 * (π/18)) = cos(π/9)
Continue evaluating the function at each left endpoint of the subintervals.
Step 5: Multiply each function value by the width of the subinterval.
Multiply the function value at each left endpoint by the width (Δx) of the corresponding subinterval.
For example:
For subinterval 1: 1 * Δx
For subinterval 2: f(π/18) * Δx
Continue multiplying the function value at each left endpoint by the width of the corresponding subinterval.
Step 6: Sum up all the subinterval areas.
Add up the area approximations for each subinterval to get the total approximate area.
Approximate Area = (Σ f(x) * Δx) for all subintervals
For example:
Approximate Area = (1 * Δx) + (f(π/18) * Δx) + ... + (f(8π/18) * Δx)
Step 7: Round your answer to four decimal places.
Finally, round the approximate area to four decimal places.
That's how you can use left endpoints and 9 rectangles to find the approximation of the area between the graph of the function cos(2x) and the x-axis over the interval [0, π/2].