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

Answer:

lateral​ surface area of the cone =49π

Step-by-step 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

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

=32π * 49/(16 *2)

=49π