**Title**: Calculating Surface Area of a Rotated Object**Problem Statement**: Calculate the surface area of the object obtained by rotating the following shaded area around the x-axis.**Graph Description**:- The graph consists of a right-angled triangle in the coordinate plane.- The triangle's base extends from x = -1 to x = 1 on the x-axis.- The height of the triangle reaches y = 1 on the y-axis.- The shaded area is the interior of this triangle.**Explanation**:The shaded triangle will be rotated around the x-axis to generate a three-dimensional object, specifically a cone. To find the surface area of this cone, calculus methods like surface area integrals can be applied. Note that the generated surface area will include only the lateral surface of the cone and not the bases.

Given the sketch of the shaded region Find the surface area of the object obtained by rotating the shaded region around x-axis.First, find the equation of the line joining (-1,0) and (0,1). y-01-0=x+10-(-1)⇒y=x+1 The equation of the line joining (0,1) and (1,0) is y-10-1=x-01-0⇒y=1-x The surface area is S=S1+S2 where S1 is the surface area of the solid obtained by rotating y=x+1, -1≤x≤0 about the x-axis and S2 is the surface area of the solid obtained by rotating y=1-x, 0≤x≤1 about the x-axis. The surface area can be found using the formula S=∫2πy ds where ds=1+dydx2dx if y=f(x), a≤x≤b.Find S1. Take the derivative of y=x+1. dydx=1 Then ds=1+12dx=2 dx. S1=∫-102πx+12 dx =22π∫-10x+1 dx =22πx22+x-10 =22π0-12-1 =2πFind S2. Take the derivative of y=1-x. dydx=-1 Then ds=1+-12dx=2 dx. S2=∫012π1-x2 dx =22π∫011-x dx =22πx-x2201 =22π1-12 =2π Hence the surface area S is S=2π+2π=22π