Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal places.)
g(x) = 7 sin x, [0, π], 6 rectangles
left endpoints approximation Area:
right endpoints approximation Area:
g(x) = 7 sin x, [0, π], 6 rectangles
left endpoints approximation Area: 13.6787
right endpoints approximation Area: 13.6787
Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal places.)
g(x) = 4 sin x, [0, π], 6 rectangles
3.73
To find the left endpoints approximation and the right endpoints approximation of the given function, you need to divide the interval [0, π] into equal subintervals (rectangles) and calculate the area of each rectangle.
Step 1: Determine the width of each rectangle.
Divide the width of the interval [0, π] by the number of rectangles, which is 6 in this case.
Width of each rectangle = (π - 0) / 6 = π / 6
Step 2: Calculate the height of each rectangle.
For the left endpoints approximation, you need to evaluate the function at the left endpoint of each subinterval.
For the right endpoints approximation, you need to evaluate the function at the right endpoint of each subinterval.
For the left endpoints, evaluate the function at x-values: 0, π/6, π/3, π/2, 2π/3, 5π/6.
Height of each rectangle = g(x)
For the right endpoints, evaluate the function at x-values: π/6, π/3, π/2, 2π/3, 5π/6, π.
Height of each rectangle = g(x)
Step 3: Calculate the area of each rectangle.
For each rectangle, multiply the width by the height.
For the left endpoints approximation:
Area = (width of rectangle) * (height of rectangle)
Area = (π/6) * (g(0) + g(π/6) + g(π/3) + g(π/2) + g(2π/3) + g(5π/6))
For the right endpoints approximation:
Area = (width of rectangle) * (height of rectangle)
Area = (π/6) * (g(π/6) + g(π/3) + g(π/2) + g(2π/3) + g(5π/6) + g(π))
Now, let's calculate the areas:
For the left endpoints approximation:
Height of each rectangle = g(0) + g(π/6) + g(π/3) + g(π/2) + g(2π/3) + g(5π/6)
Area = (π/6) * Height
For the right endpoints approximation:
Height of each rectangle = g(π/6) + g(π/3) + g(π/2) + g(2π/3) + g(5π/6) + g(π)
Area = (π/6) * Height
Now, substitute the respective x-values in the function g(x) = 7 sin x and calculate:
For the left endpoints approximation:
Height = 7sin(0) + 7sin(π/6) + 7sin(π/3) + 7sin(π/2) + 7sin(2π/3) + 7sin(5π/6)
For the right endpoints approximation:
Height = 7sin(π/6) + 7sin(π/3) + 7sin(π/2) + 7sin(2π/3) + 7sin(5π/6) + 7sin(π)
Now, use a calculator to evaluate the trigonometric functions and calculate the areas by multiplying the respective heights by the width (π/6). Round the answers to four decimal places.
clearly the rectangles are bounded at
x = 0, π/6, π/3, π/2, 2π/3, 5π/6, π
So, evaluate g(x) at the boundaries and add up the areas. What do you get?
There are plenty of online calculators for this to check your work.