Question

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?

Answer 1

Use your calculator.

Explanation:

`4 sin frac{3 pi}{12} * frac{3 pi}{12}`
`+ 4 sin frac{2 * 3 pi}{12} * frac{3 pi}{12}`
`+ 4 sin frac{3 * 3 pi}{12} * frac{3 pi}{12}`
`+ 4 sin frac{4 * 3 pi}{12} * frac{3 pi}{12}`
`+ 4 sin frac{5 * 3 pi}{12} * frac{3 pi}{12}`
`+ 4 sin frac{6 * 3 pi}{12} * frac{3 pi}{12}`

Answer 2

`R RS = 1.355`

Explanation:

We have:

`f(x) = 4sinx`

We want to calculate over the interval `[0,(3pi)/2]` with `5` 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;

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`