How does one use trapezoid rule to approximate volume of a revolution around the x axis?
The region shaded in the figure at the right is rotated about the x-axis. Using the trapezoid rule with 5 equal subdivisions, the approximate volume of the resulting solid is...the answer is 127.
The values give in the graph are:
f(1)=2, f(2)=3, f(3)=4, f(4)=3, f(5)=2, f(6)=1
and the shaded region is from x=1 to x=6.
I know how to find volume and I know how to use trapezoid rule but I have no idea how to combine them. Can anyone shed some light on the way to solve this problem? Thank you!
$\endgroup$ 12 Answers
$\begingroup$Here is the graph
The main problem here, is to find the volume of the solid of revolution of one trapezoid.
Using the shell method , where $V_{tr}=V_{(generic trapezoid)}$:
$$V_{tr}=\int_{x_1}^{x_2}\pi f^2(x)dx$$
Where $f(x)$ is the approximation for the trapezoids, and not the original f function.
Which is the area of one point of f rotated around the x axis ($f(x)=radius$), and integrating it to include all x's. In the case of the first trapezoid:
$$V_{tr1}=\pi\int_{1}^{2}(x+1)^2 dx=\pi\int_{1}^{2}(x^2+2x+1)dx=\pi \left[\frac{x^3}{3}+x^2+x\right]\biggr\rvert_{1}^{2}=\frac{19\pi}{3} $$
Do the same for the other trapezoids, but with the formula for the ones with $3<x<6$ being $f(x)=-x+7$
$\endgroup$ $\begingroup$HINT:
For each trapezium you need to find general formula for $ \bar y $ of fixed shell thickness or x- width.
$$ y = m x + c $$
$$ \bar y = \dfrac{\int y dA}{dA} = \dfrac{\int y^2 dx}{dA} $$
You can take it from there using Pappu's theorem to find rotated volumes as you know all else that is required.
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