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MrMuchimi
3 years ago
14

Use the Midpoint Rule with n = 5 to estimate the volume V obtained by rotating about the y-axis the region under the curve y = 1

+ 9x3 , 0 ≤ x ≤ 1. (Round your answer to two decimal places.)
Mathematics
1 answer:
velikii [3]3 years ago
3 0

Using the shell method, the volume is given exactly by the definite integral,

2\pi\displaystyle\int_0^1x(1+9x^3)\,\mathrm dx

Splitting up the interval [0, 1] into 5 subintervals gives the partition,

[0, 1/5], [1/5, 2/5], [2/5, 3/5], [3/5, 4/5], [4/5, 5]

with left and right endpoints, respectively, for the i-th subinterval

\ell_i=\dfrac{i-1}5

r_i=\dfrac i5

where 1\le i\le5. The midpoint of each subinterval is

m_i=\dfrac{\ell_i+r_i}2=\dfrac{2i-1}{10}

Then the Riemann sum approximating the integral above is

2\pi\displaystyle\sum_{i=1}^5m_i(1+9{m_i}^3)\frac{1-0}5

\dfrac{2\pi}5\displaystyle\sum_{i=1}^5\left(\frac{2i-1}{10}+9\left(\frac{2i-1}{10}\right)^4\right)

\dfrac{2\pi}{5\cdot10^4}\displaystyle\sum_{i=1}^5\left(16i^4-32i^3+24i^2+1992i-999\right)=\frac{112,021\pi}{25,000}\approx\boxed{14.08}

(compare to the actual value of the integral of about 14.45)

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