1.

Consider the following curve. _ 3 y x , 0 E X i: 5 Set up an integral in terms ofx that can be used to find the area of the surface 5 obtained by rotating the curve about the x-axis. s-/:[E]e D Find the exact area of the surface obtained by rotating the curve about the x-axis. The given curve is rotated about the y-axis. Set up, but do not evaluate, an integral for the area of the resulting surface by integrating (a) with respect to X and (b) with respect to y. y=5+sin(x), stsl 2 (a) Integrate with respect to X. #2 f (S i a\" O (b) Integrate with respect to y. ( )0\"? This question has severai parts that must be oompieted sequentiaiiv. If you skip a part of the question, you wili not receive any points .for the skipped part, and you wiii not be able to come back to the skipped part, Tutorial Exercise Find the exact area of the surface obtained by rotating the curve about the X-axis. We are asked to nd the surface area of the curve dened by y = x3 rotated about the x-axis over the interval 0 S x S 4. Recall the following formula for the surface area of a function ofx rotated about the Xaxis. Note that as the curve rotates in a circular manner about the xaxis, the expression 2:11,! is the circumference of radius 3/ and the radical measures the arc length that is the width of a band. .5 ' 2 S =/ 2y 'Y'l 1 + (3) dx 3 0'): We begin by substituting y = x3 4 l 2 3 = 2910(3) 1 +( ) dx i \\/ 4 = / 2?T[X3} 1 + x4 dx 0 and its derivative in the surface area formula and simplifying. Find the exact area of the Surface obtained by rotating the curve about the x-axie. The given curve i5 rotated about the y-axie. Find the area of the resulting surface. 1 2|n[x}, 35x25 F_13 4 2 E