Use midpoints to approximate the area under the curve y = f(x) = 5sin(pi x) + 2.5cos(4pi x) on the interval [0, 1] using 10 equal subdivisions.
Please give final answer thanks!
To approximate the area under the curve using midpoints, we need to divide the interval [0, 1] into 10 equal subdivisions.
First, we need to find the width of each subdivision,
Width of each subdivision = (b - a) / n
where b = 1 (upper limit of the interval), a = 0 (lower limit of the interval), and n = 10 (number of subdivisions).
Width of each subdivision = (1 - 0) / 10
Width of each subdivision = 0.1
Now, we can calculate the midpoints of each subdivision by adding half of the width to the lower limit of each subdivision.
Midpoint of the 1st subdivision = 0 + (0.1 / 2) = 0.05
Midpoint of the 2nd subdivision = 0.1 + (0.1 / 2) = 0.15
Midpoint of the 3rd subdivision = 0.2 + (0.1 / 2) = 0.25
...
And so on, until the midpoint of the 10th subdivision.
Once we have the midpoints, we can substitute these values into the equation of the curve y = 5sin(pi x) + 2.5cos(4pi x) to find the corresponding y-values.
Next, we need to calculate the area of each rectangle by multiplying the width of each subdivision by the corresponding y-value obtained from the curve.
Finally, we add up the areas of all the rectangles to approximate the total area under the curve.
To do this, we sum up the areas of each rectangle:
Approximate area = (width of 1st rectangle * height of 1st rectangle) + (width of 2nd rectangle * height of 2nd rectangle) + ... + (width of 10th rectangle * height of 10th rectangle)
By performing these calculations, you will obtain the final answer to approximate the area under the curve.