How do you use Riemann sums to evaluate the area under the curve of f(x) = 4 sin xf(x)=4sinx on the closed interval [0, 3pi/2], with n=6 rectangles using right endpoints?

2 Answers
Jul 7, 2017

Use your calculator.

Explanation:

4 sin frac{3 pi}{12} * frac{3 pi}{12}4sin3π123π12
+ 4 sin frac{2 * 3 pi}{12} * frac{3 pi}{12}+4sin23π123π12
+ 4 sin frac{3 * 3 pi}{12} * frac{3 pi}{12}+4sin33π123π12
+ 4 sin frac{4 * 3 pi}{12} * frac{3 pi}{12}+4sin43π123π12
+ 4 sin frac{5 * 3 pi}{12} * frac{3 pi}{12}+4sin53π123π12
+ 4 sin frac{6 * 3 pi}{12} * frac{3 pi}{12}+4sin63π123π12

Jul 7, 2017

R RS = 1.355 RRS=1.355

Explanation:

We have:

f(x) = 4sinx f(x)=4sinx

We want to calculate over the interval [0,(3pi)/2][0,3π2] with 55 strips; thus:

Deltax = ((3pi)/2-0)/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;

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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)/2-1)
" " = 1.1715