Question

Find the lateral​ (side) surface area of the cone generated by revolving the line segment y equals one fourth x ​, 0 less than or equals x less than or equals 7​, about the​ x-axis. The lateral surface area of the cone generated by revolving the line segment y equals one fourth x ​, 0 less than or equals x less than or equals 7​, about the​ x-axis is nothing. ​(Type an exact​ answer, using pi as​ needed.)

Answer 1

lateral​ surface area of the cone =49π

Explanation:

y= x/4

dx/dy = 4

Given x range as from 0 ≤ x ≤ 7 ⇒ x ranges between (a,b)

a= x/4 = 0/4 = 0

b = x/4= 7/4

lateral​ surface area of the cone

`\int\limits^b_a {2\pi x\sqrt{1 + ((dx)/(dy))^2 } } \, dx \\\\= \int\limits^{(7)/(4)}_0 {2\pi (4y)√(1 + (4)^2 ) } \, dx \\\\= 32\pi \int\limits^{(7)/(4)}_0 { y } \, dx \\\\= 32\pi [(y^2)/(2)]|^(7)/(4)_0`

=32π * 49/(16 *2)

=49π