How do you use Riemann sums to evaluate the area under the curve of # f(x) = 4 sin x# on the closed interval [0, 3pi/2], with n=6 rectangles using right endpoints?
# R RS = 1.355 #
We have:
# f(x) = 4sinx #
We want to calculate over the interval
# Deltax = ((3pi)/20)/6 = (3pi)/12#
Note that we have a fixed interval (strictly speaking a Riemann sum can have a varying sized partition width). The values of the function are tabulated as follows;
Right Riemann Sum
# R RS = sum_(r=1)^4 f(x_i)Deltax_i #
# " " = 0.1309 * (0.5221 + 1.0353 + 1.5307 + 2 + 2.435 + 2.8284)#
# " " = 0.1309 * (10.3516)#
# " " = 1.355 #
Actual Value
For comparison of accuracy:
# Area = int_0^((3pi)/12) \ 4sinx \ dx #
# " " = [4cosx]_0^((3pi)/12) #
# " " = 4[cosx]_0^((3pi)/12) #
# " " = 4(cos ((3pi)/12)cos0) #
# " " = 4(sqrt(2)/21)#
# " " = 1.1715#
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To use Riemann sums to evaluate the area under the curve of ( f(x) = 4 \sin x ) on the closed interval ( [0, \frac{3\pi}{2}] ) with ( n = 6 ) rectangles using right endpoints, follow these steps:

Determine the width of each rectangle: ( \Delta x = \frac{b  a}{n} ), where ( a = 0 ), ( b = \frac{3\pi}{2} ), and ( n = 6 ).

Calculate the right endpoints for each subinterval: ( x_i = a + i \cdot \Delta x ) for ( i = 1, 2, \ldots, n ).

Evaluate the function ( f(x) = 4 \sin x ) at each right endpoint to find the heights of the rectangles: ( f(x_i) ) for ( i = 1, 2, \ldots, n ).

Multiply the height of each rectangle by its width to find the area of each rectangle.

Sum up the areas of all rectangles to get the Riemann sum: ( \sum_{i=1}^{n} f(x_i) \Delta x ).

Finally, take the limit of the Riemann sum as ( n ) approaches infinity to find the exact area under the curve.
By following these steps with ( n = 6 ), you can approximate the area under the curve of ( f(x) = 4 \sin x ) on the interval ( [0, \frac{3\pi}{2}] ).
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