Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 5 36 − x2 , y = 0, x = 2, x = 4; about the x-axis

Answer 1

Answer: V = 193.25π

Step-by-step explanation: The method to calculate volume of a solid of revolution is given by an integral of the form:

V =  $\pi\int\limits^a_b {[f(x)]^{2}} \, dx$

f(x) is the area is the function that rotated forms the solid.

For f(x)=y= $\frac{5}{36}-x^{2}$ and solid delimited by x = 2 and x = 4:

V = $\pi\int\limits^4_2 {(\frac{5}{36}-x^{2} )^{2}} \, dx$

V = $\pi\int\limits^4_2 {(\frac{25}{1296}-\frac{10x^{2}}{36}+x^{4}) } \, dx$

V = $\pi(\frac{25.4}{1296}-\frac{10.4^{3}}{108}+\frac{4^{5}}{5}-\frac{25.2}{1296}+\frac{10.2^{3}}{108}-\frac{2^{5}}{5} )$

V = $\pi(\frac{50}{1296}-\frac{560}{1296}+\frac{992}{1296} )$

V = 193.25π

The volume of a solid formed by y = $\frac{5}{36} - x^{2}$ and delimited by x = 2 and x = 4

is 193.25π cubic units.